Elongation and shape changes in organisms with cell walls: a dialogue between experiments and models
EElongation and shape changes in organisms with cell walls: adialogue between experiments and models
Jean-Daniel Julien a,b , Arezki Boudaoud b, ∗ a Laboratoire Reproduction et D´eveloppement des Plantes, Universit´e de Lyon, ENS de Lyon, UCB Lyon 1, CNRS,INRA, 46 all´ee d’Italie, 69364 Lyon Cedex 07, France b Laboratoire de Physique, Univ. Lyon, ENS de Lyon, UCB Lyon 1, CNRS, 46 all´ee d’Italie, 69364 Lyon Cedex 07,France
Abstract
The generation of anisotropic shapes occurs during morphogenesis of almost all organisms. Withthe recent renewal of the interest in mechanical aspects of morphogenesis, it has become clearthat mechanics contributes to anisotropic forms in a subtle interaction with various molecularactors. Here, we consider plants, fungi, oomycetes, and bacteria, and we review the mechanismsby which elongated shapes are generated and maintained. We focus on theoretical models of theinterplay between growth and mechanics, in relation with experimental data, and discuss howmodels may help us improve our understanding of the underlying biological mechanisms.
Keywords: morphogenesis, cell wall, bacteria, fungi, yeasts, oomycetes, plants, symmetrybreaking, cell polarity
1. Introduction
Symmetry breaking is a fascinating feature of morphogenesis: How does a sphere-like or-ganism become rod-like? How does a rod-like organism maintain its shape? Symmetry break-ing occurs within cells, when cell polarity is established and proteins or organelles are asy-metrically distributed in the cytoplasm or at the plasma membrane (Bornens, 2008; Goehringand Grill, 2013). Cell polarity has been extensively investigated, notably using modelling ap-proaches (Mogilner et al., 2012). In this review, we consider symmetry breaking when it isassociated with shape changes, as exemplified by branching during hydra development (Merckeret al., 2015). More generally, symmetry breaking occurs when an initially symmetric shape —e.g. a sphere that is unchanged by rotations around its centre, or a cylinder that is unchangedby rotations around its axis — or an initially symmetric distribution of molecules (e.g. homo-geneously distributed on a sphere) becomes asymmetric — a bump appears on the sphere or onthe cylinder, or the molecules become more concentrated around a point of the sphere. Here wefocus on walled cells and particularly consider plants, fungi, oomycetes, and bacteria. The cellsof these organisms are characterised by a relatively high internal hydrostatic pressure known asturgor pressure, which results from the concentration of solutes in the cytosol, and drives growth ∗ Email address: [email protected] (Arezki Boudaoud)
Preprint submitted to The Cell Surface April 19, 2018 a r X i v : . [ q - b i o . CB ] A p r though this is debated, see e.g. Harold et al., 1996; Rojas et al., 2014). Turgor pressure is coun-terbalanced by a rigid shell, the extra-cellular matrix known as the cell wall, that prevents cellsfrom bursting (Harold, 2002). As such, the cell wall is fundamental in determining cell shape. Al-though the cell walls of these organisms di ff er in their chemical composition (Lipke and Ovalle,1998; Cosgrove, 2005; Silhavy et al., 2010), they have similar mechanical properties that are reg-ulated so as to shape cells. In particular, pressure being a global and isotropic (non-directional)force, inhomogeneous or anisotropic distributions of mechanical or biochemical properties seemneeded to establish and maintain asymmetric shapes.Previous reviews addressed such questions for specific systems, see for instance (Dav`ı andMinc, 2015, for fission yeast), (Chang and Huang, 2014, for bacteria and fission yeast), (Kroegerand Geitmann, 2012, for pollen tubes), or (Uyttewaal et al., 2010, for plants). In this review, wediscuss the links between mechanics and growth of elongated cells across kingdoms. We payspecific attention to computational modelling, because it appears as a powerful tool to validatehypotheses by setting aside all but the fundamental actors of the phenomena investigated and toguide future experimental e ff ort. We apologise to those whose work could not be included.
2. Systems of interest and their cell walls
We first present a few model systems; we give a brief introduction to their cell walls interms of composition, structure, and mechanical properties, along with a presentation of availablemechanical models of cell walls.
All kinds of shapes are found amongst bacteria. Actually, shape has long been an impor-tant criterion for classification (Cabeen and Jacobs-Wagner, 2005) and is fundamental for manybacterial functions (Young, 2006; Singh and Montgomery, 2011; Chang and Huang, 2014). Therod is a very common shape, with
Escherichia coli as a representative Gram-negative species,making bacteria good systems to study the establishment and maintenance of elongated shapes.Many rod-shaped bacteria expand di ff usively, with new cell wall incorporated all along the rod,while other expand at a pole or within a restricted region Cava et al. (2013).The typical bacterial cell wall is mostly made of peptides and glycans (Cabeen and Jacobs-Wagner, 2007; Silhavy et al., 2010). The glycan strands are oriented circumferentially and givethe cell wall anisotropic mechanical properties (Chang and Huang, 2014), the cell wall beingsti ff er in the circumferential direction than in the longitudinal direction. In Gram-positive bac-teria, the structure of the cell wall is not as well characterised as in Gram-negative bacteria, al-though recent studies also support a circumferential arrangement of glycan strands (Beeby et al.,2013). A detailed mechanical model of the cell wall of Gram-negative bacteria was proposedby Huang et al. (2008), with the peptidoglycan network simulated like a network of springs; byadding defects in the network and modulating the type of defects and their density, this modelwas able to reproduce several bacterial cell morphologies: curved, helical, snake-like, and lemonshapes. Whereas this model did not take growth into account, it was the starting point for manymodels of wall expansion, which are discussed in the next section. Many fungi have elongated cells that grow from their tips: hyphae and yeasts. Hyphaeare very long filamentous cells, that can be collectively organised into a mycelium. Yeasts are2 uter membraneInner membrane Peptidoglycan network A. Rod-shaped bacterium
Plasma membrane MannanGlucanChitin (Towardmycelium) B. Fungi: hypha and fission yeast
Plasma membraneCellulose/hemi-cellulosenetwork + pectin matrixCallose (pollen tube) (Shoot) C. Root hair, pollen tube and SAM (Toward root or pollen grain)
Figure 1:
Systems of interest: growth mode and composition of the cell wall. (A) In many rod-shaped bacteria,such as
E. coli , growth is di ff use, localised on the whole cylindrical part on the cell. The cell wall of E. coli and otherGram-negative bacteria is composed of a sti ff layer of peptides and glycans surrounded by two lipid membranes (Silhavyet al., 2010; Chang and Huang, 2014). (B) Tip growth is observed in fungal hyphae and fission yeast. According tomodels, the cell wall would be composed of three layers, made respectively of chitin, glucan and mannan (Lipke andOvalle, 1998; Bowman and Free, 2006). (C) In plants, pollen tubes and root hairs are tip-growing cells. On a largerscale, growth is focused on the tip of emerging organs around the shoot apical meristem. The plant cell wall is made ofa network of cellulose embedded in a matrix of hemicellulose and pectin (Cosgrove, 2005). Callose can also be found,notably in pollen tubes (Chebli and Geitmann, 2011). In the three systems, the relative amounts of components may vary,for instance between species or even spatially within a single cell. Schizosaccharomyces pombe ) is a good systemto study cell polarity and the maintenance of rod shapes (Chang and Martin, 2009). Indeed,it grows as a capped cylinder, maintaining a constant diameter (except for spores, which areroughly spherical).The cell wall of fungi is mostly made of glucans (excluding cellulose), mannoproteins, andchitin (Lipke and Ovalle, 1998). Although many of these components are fibrous, it is believedthat the fungal cell wall does not have anisotropic mechanical properties because of the lack ofpreferential orientation of the fibres (Chang and Huang, 2014). (Strictly speaking, the fibres aremostly tangential, and the cell wall is transversely isotropic, being softer across the thicknessthan in the directions tangential to the wall). Despite a rather well-known composition, fungalcell walls have not been modelled in detail.Although oomycetes grow in mycelial forms like fungi, they belong to a di ff erent taxonomicgroup, Stramenopiles. Their cell walls are mostly made of glucans and, unlike fungi, they containsome cellulose and tiny amounts of chitin (M´elida et al., 2013). Plants provide two other systems of interest, pollen tubes and root hairs, that elongate throughtip growth. The pollen tube is a long protuberance that grows out from the pollen grain until itreaches the ovule for fertilisation (Geitmann, 2010; Kroeger and Geitmann, 2012). Accordingly,its growth is fast and highly directional. Its growing tip is formed of a single cell. Root hairsare long tubular outgrowths from specialised epidermal cells of the root. They are important forabsorption of nutrients and anchorage to the soil (Carol and Dolan, 2002; Grierson et al., 2014).The plant cell wall is mostly made of polysaccharides: cellulose microfibrils embedded ina matrix of hemicelluloses and pectins (Varner and Lin, 1989). Cellulose fibrils can be muchlonger than the cell diameter and their organisation di ff ers according to cell type and to develop-mental stage, ranging from highly directional circumferential alignment to random orientations(Cosgrove, 2005). As cellulose is the sti ff est component of the wall, a preferential orientationof microfibrils can give anisotropic properties to the plant cell wall: It is sti ff er in the directionof the fibres (Kerstens et al., 2001), which may lead to less expansion in this direction, and sodrive anisotropic cell growth. In pollen tubes and root hairs, cellulose usually displays a helicalarrangement that could help resist bending forces and penetrate external medium (Aouar et al.,2010); this arrangement could also reinforce the transition region between the tip and the cylin-drical part, which bears the highest tension (Geitmann, 2010). An early model of the cell wallfocused on the self-organisation of cellulose due to cell geometry (Emons and Mulder, 1998;Mulder and Emons, 2001); the condition of optimal packing of cellulose microfibrils restrainstheir direction and the movement of their synthesising complexes along the axis of the cell cangenerate various types of organisations with locally aligned fibres. More recently, models at-tempted to define realistic geometries for the arrangement of polysaccharides in the cell wall andto predict the corresponding elastic properties for small deformations (Qian et al., 2010; Khaet al., 2010; Yi and Puri, 2012). Some of the relevant results that we will discuss were obtained in a multicellular context.The shoot apical meristem (SAM) is the tissue located at the tip of any above-ground branch ina plant; it contains a stem cell pool and is the site of organogenesis (Ha et al., 2010; Murrayet al., 2012; Gaillochet et al., 2014). Organs are initiated around the tip and emerge as protuber-ances from the apical dome, breaking the symmetry around the axis of the dome. As discussed4bove, the deposition of cellulose microfibrils in a preferential direction can provide anisotropicmechanical properties to the SAM.
3. Generating elongated shapes
We first consider geometrical models built in the context of hyphal growth. In hyphae ofmany fungi, a intriguing structure localised close to the tip and known as the spitzenk¨orper (SPK)concentrates vesicles and is thought to be the organising centre of tip growth. It has been pro-posed that vesicles, containing notably multiple cell-wall regulating enzymes, are transported tothe region of the SPK by cytoplasmic microtubules. Actin microfilaments, found in the SPK,then take over in regulating the supply of enzymes and material to the membrane. The com-plete composition of the SPK is still unclear (see Steinberg et al., 2017, for a review on hyphalgrowth). Its description as a cluster of cell wall-building enzymes has led to the Vesicle SupplyCentre (VSC) model (Bartnicki-Garcia et al., 1989) of fungal growth, first implemented in twodimensions. In this framework, the VSC is a point in space that constantly emits vesicles inrandom directions. Those vesicles move at constant velocity, and locally increase the length ofthe cell wall after they have reached it. Finally, the VSC moves at a constant, prescribed veloc-ity. This yields a steady shape that compares well with experimental observations of growinghyphae. The three-dimensional generalisation of the VSC model (Gierz and Bartnicki-Garcia,2001) raised the question of how the material brought by the vesicle is distributed between thelongitudinal and the circumferential directions, which led to propose additional rules for expan-sion. A second improvement of the VSC model was to replace the ballistic motion of the vesiclesby di ff usion (Tindemans et al., 2006), which only slightly modifies the shapes generated by themodel. VSC models were successful in demonstrating that self-similar tip growth can emergefrom patterns of exocytosis. However, they assumed that wall expansion is limited by supply ofmaterials, and that cell wall mechanics is negligible. While there is experimental evidence thatexocytosis is required for growth, it is also clear that cell wall mechanics is important to set thepace of expansion (Kroeger and Geitmann, 2012).Cell wall mechanics was accounted for in several generic models. A first class of modelsassumes that growth can be considered as a viscous process, whereby the cell wall expands likea viscous material under the tension generated by turgor pressure; such a process would leadto wall thinning, and so cell wall synthesis is assumed to maintain the cell wall thickness at anapproximately constant value. Bernal et al. (2007) got inspiration from the inflation of rubberballoons, which were modelled as elastic shells with spatially varying sti ff ness: More precisely,the compliance of the material (how easy to stretch it is) is large on a narrow annular regionaround the tip and small on the cylindrical part of the shell. This model was able to reproduceobserved deformations of root hairs. This work highlighted the importance of modulating theglobal pressure drive by local supply of cell wall material, or by local modifications of cell wallproperties. The spatial extent of the wall deposition, and more precisely how it depends on thesize of the cell, was theoretically found to change the shape of the growing tip (Camp`as andMahadevan, 2009). In this latter study, the cell wall was modelled as a thin viscous shell withinfinite viscosity on the flanks (so that the tube maintains its diameter). Growth is compensatedby material addition (synthesis) at the tip. The authors investigated the dependence of cell shapeon viscosity and spatial extent of material addition; they showed that tube radius increases withviscosity and that the tip becomes more blunt as the spatial extent is increased. These results5rompted a broad analysis of tip-growing cell shape from various species (plants, fungi andoomycetes; Camps et al., 2012): The tip radius (radius of curvature at the tip) scales with tuberadius in plants and fungi, while the tip radius appears constant in oomycetes.A second class of models considers growth as an incremental process whereby, at each step,the cell wall is elastically stretched by turgor pressure, and this stretched configuration is takenas the starting point of the next step; again, synthesis is assumed to keep cell wall thicknessconstant. Goriely and Tabor (2003) used the framework of the nonlinear elasticity theory of thinshells, which allows for large deformations of the shells. An important ingredient of their modelis that the tip of the hypha is softer than the cylindrical part. Consistent with the incrementalframework, growth is simulated by computing the deformation of the shell due to turgor pressurethen taking the deformed shape as a new initial shape that can be deformed further. So in thismodel cell wall expansion is localised due to the softer tip. More recently, this model wasused to study the e ff ect of the friction between the growing tip and the external medium. Thisfriction leads to a flattening of the tip that is consistent with experimental data (Goriely andTabor, 2008). It can be shown that incremental models are mathematically equivalent to viscousmodels in the limit where pressure is relatively small (see for instance Bonazzi et al., 2014);nevertheless, the interpretation of parameters di ff ers between the two types of model. In viscousmodels, the viscosity (inverse of extensibility) is a proxy for the rate of cell wall remodellingunder tension, but this viscosity cannot be measured directly in experiments. In elastic models,the elastic modulus (inverse of compliance) quantifies the sti ff ness of the wall material, whichcan be measured experimentally, but it does not necessarily predict how fast the cell wall expandsunder tension. Actually, the chemistry of the cell wall is an important ingredient that is missingfrom these two types of models, a limitation that applies to most of the mechanical modelspresented here.In plants, root hairs and pollen tubes are quite similar to fungi with respect to the controlof polar growth. Microtubules have two types of localisation in plants: cortical – close to theplasma membrane, and endoplasmic. In root hairs and pollen tubes, microtubules, as well asactin filaments, are oriented along the tube axis and are involved in targeting the supply of newmaterial to the tip (Sieberer et al., 2005; Gu and Nielsen, 2013; Chebli et al., 2013). In roothairs, this organisation of microtubules depends on the cell nucleus (Ambrose and Wasteneys,2014). Detailed measurements in root hairs have shown that cell wall expansion occurs mainlyin an annulus just behind the tip, and is isotropic there; farther from the tip, expansion becomesmostly radial and decays with distance to the tip (Shaw et al., 2000). Dumais et al. (2006) builta mechanical model that qualitatively reproduced expansion profiles in several tip growing cells,including root hairs, using the following assumptions. The cell wall is viscoplastic: expansionoccurs above a threshold in tension and then increases linearly with tension; the viscosity (inverseof extensibility) is smaller close to the tip. The thinning of the wall due to its stretching iscompensated by deposition so as to keep its thickness constant. The main result is that themodel accounts for quantitative measurements of cell geometry (curvatures) and wall expansion(strain rates) in root hairs only if cell walls have mechanical anisotropy. More precisely, the cellwall needs to be transversely isotropic, meaning that its properties in the direction of thicknessare di ff erent from its properties in its tangent plane. Such anisotropy can be explained by thedeposition of cellulose tangentially to the cell wall.A similar pattern of expansion is observed in pollen tubes (Zerzour et al., 2009; Hepler et al.,2013). The shape of tubes was reproduced with an incremental elastic model that included asharp gradient of sti ff ness at the tip (Fayant et al., 2010). The best fitting of observed shapeswas achieved assuming the cell wall transversely isotropic. Interestingly, the gradient of sti ff ness6sed in the simulation is consistent with the gradients of density observed for various cell wallcomponents, such as pectins, cellulose, and callose, that determine its mechanical properties.As a complement to the systems studied here, we also mention trichomes — elongated haircells found in the aerial part of plants — because they appear to di ff er from pollen tubes. Yanag-isawa et al. (2015) used a viscoelastic thin-shell model of the cell wall, expanding due to turgor-generated tension. They needed to combine softer tip and mechanical anisotropy on the sides tobetter match experimental data.In fungi, the local delivery of new cell wall is driven by microtubules and actin filaments.The cytoskeleton could be directly required for growth or only define the location where wall ex-pansion takes place. Chemical treatments and mutants have demonstrated that disruption of mi-crotubules leads to major geometrical defects in the fission yeast (Hagan, 1998). Application ofactin inhibitors can modify the dynamics of growth or completely arrest it depending on the con-centrations used. Consequently, the microtubules and actin filaments must e ffi ciently target thecell tips (Sawin and Nurse, 1998; Terenna et al., 2008) to deliver the new material to the properlocation. Drake and Vavylonis (2013) modelled the coupling between microtubule dynamics, aremodelling signal – a protein required for cell wall remodelling, and cell wall mechanics. As inmany previous studies, they considered the cell wall as e ff ectively viscous (Camp`as and Mahade-van, 2009) and they assumed in addition that e ff ective viscosity is reduced by the remodellingsignal. They considered microtubules as growing and shrinking flexible rods attached to the nu-cleus. They first assumed that the level of the remodelling signal is imposed by the likelihoodof contact between microtubules and cell wall, but they found that this did not enable the main-tenance of cell width over many generations (as observed in living cells). Maintenance of cellwidth was achieved with additional assumptions: microtubules control the deposition of land-mark proteins that in turn attract the remodelling signal; the remodelling signal has an intrinsicdynamics (such as reaction-di ff usion) that leads to its localisation over a region of well-definedsize. To summarise, Drake and Vavylonis (2013) built one of the first successful models for cellmorphogenesis that integrates cell polarity and cell wall mechanics. It would be interesting tofurther probe this model by, for instance, investigating the recovery from spheroplasts (cells thatbecame round following wall digestion) to rod-like shapes.Abenza et al. (2015) combined experiments and mechanical models to explore which cellularprocesses among polarity, exocytosis, or wall synthesis determine the pattern of cell wall expan-sion in fission yeast. They used an incremental elastic model and assumed the elastic modulusto be a function of either of the cell-end localised factors involved in the three previous cellu-lar processes. They found that exocytosis factors better predicted the observed pattern of wallexpansion. The pattern of supply of wall materials thus appears to be essential for shape, mak-ing the connection between the concepts behind mechanical models and those behind VSC-likemodels.Overall, the qualitative agreement with experiments of a range of mechanical models stronglysupports the notion that a softer / more compliant tip is required for tip growth. However it isyet di ffi cult to ascertain which models are more relevant to actual cells. Further quantitativemeasurements of wall expansion and wall mechanics are required to make further progress.It is generally believed that the epidermis of aerial plant tissues is under tension (see Petersand Tomos, 1996, for a review), which would occur for instance if the epidermis is much sti ff erthan internal tissues (as inferred in Beauzamy et al., 2015). Consequently, a tissue like the shootapical meristem (SAM) behaves mechanically like a pressurised shell, in which the epidermisplays the role of the shell, while inner layers corresponds to a liquid under pressure. The SAM istherefore comparable mechanically to the unicellular systems considered so far. Quantification7f cell wall expansion in an Arabidopsis mutant that does not produce organs ( pin-formed 1 ,defective in a protein that enables e ffl ux from cells of the phytohormone auxin) revealed higherexpansion rate in an annulus that surrounds the tip Kwiatkowska (2004), consistent with sti ff ercell walls at the tip (Milani et al., 2011, 2014) and reminiscent of expansion patterns in roothairs. However, this analogy is only partial because the mechanical properties of cell walls arelikely anisotropic in the shoot apex (see following section). Nevertheless, spatial variations in themechanical properties of cell walls seem to be required to establish the patterns of growth thatunderlie morphogenesis. For instance, the appearance of a new growth axis – the primordiumof a lateral organ such as a leaf or a flower – on the side of the meristem is associated with alocally softer cell wall (Peaucelle et al., 2011; Kierzkowski et al., 2012). This outgrowth requiresan increase in pectin demethylesterification (Peaucelle et al., 2011), which occurs in internal cellwalls before in surface walls, and is dependent on the accumulation of auxin (Braybrook andPeaucelle, 2013). Note that this constitutes an important di ff erence with pollen tubes, wherepectin demethylesterification rigidifies the cell wall and therefore inhibits growth (see Bosch andHepler, 2006, for more details). The three-dimensional patterns in cell wall properties promptedBoudon et al. (2015) to develop realistic mechanical models of tissues. Each cell wall is con-sidered as a thin surface with elastic, plastic and viscous properties. By fine tuning the sti ff nessand / or viscosity of walls, Boudon et al. (2015) were able to make one or several organs emerge.This work shows how heterogeneity in sti ff ness may generate complex shapes. Interestingly,several solutions are sometimes possible for creating a given shape. For this reason, the com-parison of computational outputs with real tissues cannot be limited to shape and requires otherexperimental observations. ff use growth The sti ff ness of a material is not just a number. The material can be anisotropic, i.e. itcan have di ff erent values of sti ff ness in di ff erent directions, like for instance a fibre-reinforcedmaterial, which is harder to stretch in the direction of the fibres. In many cases, elongation ofwalled cells requires such anisotropy.In many rod-shaped bacteria, growth occurs on the cylindrical region of the cell (Cava et al.,2013). New material is inserted as small patches on the cylindrical part of the cell, a processcoordinated by MreB filaments (Chang and Huang, 2014).Despite the growth being distributed on their cylindrical part, bacteria grow as elongatedcells. This is often thought to be caused by anisotropic reinforcement of the cell wall, eitherdirectly through a mechanical anisotropy of their material (Yao et al., 1999) or indirectly throughthe bacterial cytoskeleton itself. Jiang et al. (2011) focused on the e ff ect of MreB and builta model of elongation that reproduced shapes, divisions, and bulging in wild-type and mutantstrains of E. coli . The cell wall was considered as a continuous transversely isotropic materialwhose growth is driven by the transformation of intracellular energy into a mechanochemicalenergy, that combines the elastic energy of the cell wall and the chemical energy of new bonds.Growth is much slower than MreB dynamics, thus the mechanical e ff ect of helical MreB fila-ments is averaged over time and modelled as a radial force resisting turgor pressure and yieldinga preferred radius for the cell. Without this force, cells grew spherically. Whereas MreB maybend liposomes (Hussain et al., 2018), it is unkown whether MreB is strong enough to inducecurvature of the cell wall as it is synthesised.Banerjee et al. (2016) extended this model to alsoreproduce vibrio shape, where the rod-shaped bacterium is slightly bent (e.g. Caulobacter cres-centus ), by incorporating a preferred curvature of Crescentin-like proteins (with the same caveat8s for MreB). It would be interesting to know whether observed cell shapes would still be re-trieved by these models if forces from the cytoskeleton were replaced by anisotropy in elasticenergy or in bond energy.Starting from a previous static model of the cell wall in Gram-negative bacteria (Huang et al.,2008, see above), Furchtgott et al. (2011) modelled wall expansion by considering the insertionof new short glycan strands into existing peptide cross-links, according to one of 3 scenarii:the first scenario — random choice of the peptides — leads to bulging and loss of straightness;the two other scenarii — uniform insertion by choosing peptides inversely proportionally to theirdensity or helical insertion according to the motion of synthase — enables the maintenance of rodshapes. In order to study the molecular details of cell wall remodelling, Nguyen et al. (2015) laterbuilt a model on a similar, coarse-grained scale; they used more realistic mechanical parametersfor the peptidoglycan network, which they inferred from molecular dynamics simulations. Theyachieved maintenance of rod shapes by accounting for the spatiotemporal dynamics of enzymesinvolved in cell wall remodelling and assuming coordination between enzyme activites. Thus,the use of di ff erent microscopic hypotheses (Furchtgott et al., 2011; Nguyen et al., 2015) yieldsdi ff erent macroscopic outcomes, which calls for further comparison with experimental data. Ina more abstract model based on continuum mechanics, Amir and Nelson (2012) represented theinsertion of new material as the creation and movement of defects (dislocations in this case)along the peptidoglycan lattice. Using biological relevant values of parameters, they retrievedexponential growth in length, with a rate that is sensitive to turgor pressure. Altogether, thesestudies show that circumferential insertion of glycans is a key ingredient for maintenance ofshape.In bacteria, it is unclear whether the mechanical anisotropy of the cell wall is required forrod shapes, as direct experimental evidence is lacking while models with defects moving inisotropic walls (Amir and Nelson, 2012) may produce rod-shaped cells. In contrast, in plants,reduced cellulose content induces more isotropic growth (Baskin, 2001), supporting the ideathat mechanical anisotropy is required for anisotropic di ff use growth. In the multicellular con-text of the shoot apex, it is likely that sti ff ness anisotropy combines with the local softeningdiscussed earlier. Indeed, oriented deposition of cellulose may lead to strongly anisotropic cellwalls (Cosgrove, 2005). This would be the case in the boundary between an emerging organ andthe meristem, around the tip of the shoot, and around the tip of an organ primordium (Hamantet al., 2008), where oriented cellulose deposition is predicted based on the orientation of micro-tubules. The associated mechanical anisotropy would provide an additional mechanism for theelongation of the shoot or of the organ. Indeed, using a cell-based mechanical model of tissuegrowth, Boudon et al. (2015) and Sassi et al. (2014) showed that combining changes in sti ff nesslevels and in sti ff ness anisotropy ensures optimal outgrowth and better accounts for experimentalobservations.
4. Feedbacks that stabilise elongated shapes
An extreme case of curvature-sensing is when wall expansion is fully determined by its cur-vature. In tip growing cells, expansion is maximal at the tip, where the wall curvature is thehighest. Goriely et al. (2005) and Ja ff ar and Davidson (2013) built geometrical models of tip-growing cells in which local wall expansion (and accordingly material supply) is an increasingfunction of local curvature (Gauss curvature in 3D models). Ja ff ar and Davidson (2013) used9wo fitting parameters to reproduce the geometry of cells of many organisms (plants, fungi, acti-nobacteria), though the biological relevance of these parameters is unclear. Such models showthat curvature-based expansion is su ffi cient to account for the stable form of tip-growing cells.There is evidence for curvature-sensing in bacteria. Ursell et al. (2014) and Billings et al.(2014) found that MreB localises preferentially to regions where the wall has negative curvature;the spatiotemporal correlation of expansion and MreB indicates that MreB precedes expansion.This causal link between curvature and MreB was confirmed by perturbation experiments. Al-together, experimental data suggest that MreB relocalises to regions of negative curvature, in-ducing more expansion and reverting the local geometry to cylindrical. Ursell et al. (2014) usedthe model for glycan strand insertion (Furchtgott et al., 2011, see above) to test this mechanismin E. coli . They assumed that insertion occurs preferentially at regions of low curvature andfound that this stabilised growth of a rod-shaped cell and enabled the recovery from an initiallybent shape. Hussain et al. (2018) gave further experimental support to these conclusions bymanipulating shapes of
Bacillus subtilis by chemical treatments or by confinement in channels.However, Wong et al. (2017) combined experimental and theoretical approaches to show that theobserved local enrichment in MreB in deformed bacteria is not su ffi cient to explain the recoveryof a straight shape. Bacteria are known to sense their mechanical state through channels that are sensitive tomembrane tension. In
E. coli , Wong et al. (2017) proposed that strain-activated growth couldqualitatively explain the response and recovery of experimentally bent bacteria. Interestingly,although the biochemical implementation of this mechanism is unclear, it was suggested to workjointly with MreB-mediated regulation (as discussed above).In fission yeast, spores grow roughly spherically, until an outgrowth initiates the rod-likeshape of vegetative cells. The associated transition from unstable to stable polarity is triggeredmechanically by the rupture of the outer wall of the spore, which is a very sti ff thin layer thatsurrounds the vegetative-like wall (Bonazzi et al., 2014). Interestingly, the ratio between thevolume at the transition and the initial volume of the spore is constant, despite a large variabilityin initial volume. Bonazzi et al. (2014) modelled the outer wall as an elastic sti ff shell (that mayrupture) and considered the vegetative wall either as elastic or as viscoplastic — the viscoplasticregion is where growth occurs and it corresponds to the location of the polarisome (the ensem-ble of polarly localised proteins). The polarisome was assumed to move randomly, mimickingexperimental observations. When it is intact, the outer shell prevents outgrowth and keeps thespore roughly spherical. The tension in the shell increases, up to a threshold over which its localrupture initiates the outgrowth. The theoretical results thus show that stress-sensing via mechan-ical rupture enables the outgrowth to occur when the ratio of spore volume to initial volume hasreached a well-defined threshold.As mentioned above, morphogenesis at the shoot apex relies both on local softening andmechanical anisotropy. The two mechanisms are tuned by a feedback from mechanics. Thelocal softening that initiates organ emergence is triggered by the local accumulation of the phy-tohormone auxin (Sassi and Vernoux, 2013). Auxin patterns are determined by the polarityof PIN FORMED1 (PIN1), a membrane-addressed protein that facilitates auxin e ffl ux. Threehypotheses have been proposed for the determination of auxin polarity based on experimentalobservations (Abley et al., 2013; Sassi and Vernoux, 2013): flux of auxin, gradient of auxin,and intrinsic property of the cell that may be oriented by external cues. We here focus on the10 . Curvature-sensing in bacteria B. Growth- and force-sensingin fungi and pollen tubes C. Force-sensing in the shoot apex
Figure 2:
Feedbacks that stabilise elongation. (A) In
E. coli , the insertion of new cell wall is increased in the region ofnegative curvature. This feedback may stabilise rod shape and enable recovery from initially curved shapes (Ursell et al.,2014; Billings et al., 2014; Hussain et al., 2018). (B) In pollen tubes and fission yeast, surface expansion feeds back onmaterial supply. In pollen tubes this feedback may lead to oscillatory tip growth (Rojas et al., 2011). In fission yeast,this feedback occurs through the position of the polar cap (where polarity proteins are localised). It leads to the randomshu ffl ing of polar cap in spores. After the rupture of the outer spore wall, the feedback promotes tip-growth (Bonazziet al., 2014). (C) In the plant shoot apex, two loops involving mechanosensation are coupled. By enhancing transport ofthe phytohormone auxin, mechanical strain and stress focus growth at the tip of the organ. Mechanical stress may alsoincrease mechanical anisotropy of the cell wall via deposition of cellulose oriented by the response of microtubules tomechanical stress (Hamant et al., 2008; Heisler et al., 2010). / strain (which of strain or stress is sensed is still unclear), which is shared with the cell withhighest auxin concentration due to the induced softening of its walls. This chemomechanicalmodel has been implemented using the finite element method for the mechanics coupled with asystem of di ff erential equation for the auxin dynamics (Heisler et al., 2010). It is able to generatepatterns of auxin accumulation and to reproduce the radial PIN1 reorientation observed around acell ablation.In many plant tissues, mechanical signals also regulate the orientation of cellulose microfib-rils (Castle, 1937; Green and King, 1966; Preston, 1988; Wasteneys and Williamson, 1987, 1989;Williamson, 1990; Fischer and Schopfer, 1997; Hejnowicz et al., 2000; Hamant et al., 2008;Jacques et al., 2013; Sampathkumar et al., 2014). Indeed, cellulose is synthesised following theorientation of cortical microtubules (Baskin, 2001; Bringmann et al., 2012). Additionally, micro-tubules orient along the direction of maximal mechanical tension (Wasteneys and Williamson,1987, 1989; Williamson, 1990; Fischer and Schopfer, 1997; Hejnowicz et al., 2000; Hamantet al., 2008; Jacques et al., 2013; Sampathkumar et al., 2014). Consequently, the preferentialorientation of cellulose microfibrils reinforces the cell wall in the direction of mechanical stress(Landrein and Hamant, 2013). Several theoretical studies investigated the consequences of thisfeedback loop between mechanical stress and cell wall anisotropy. Hamant et al. (2008) mod-elled the shoot apical meristem as an elastic surface in 3D, thus only accounting for the epidermalcell layer. They used a vertex model, meaning that they only considered cell walls orthogonal tothe surface of epidermis, represented as 1-dimensional springs. The sti ff nesses of these springsincreases as they are more parallel to the local mechanical tension, mimicking the orientation ofthe microtubules and their feedback on the mechanical properties of cell walls. Growth is drivenby the turgor pressure of internal tissues. Above a stress threshold, the cell walls yield and thusdeform plastically. By initiating the emergence of an organ via the local softening of a group ofcells, the cellulose reorientation leads to a circumferential pattern around the organ, reinforcingthe boundary with the apical dome and thus making the symmetry breaking more e ff ective. Bo-zorg et al. (2014) obtained similar results using a continuous model for the epidermal layer. Theyfurther showed that using mechanical strain instead of stress as a directional cue is not su ffi cientto account for experimental observations. In fission yeast, an additional result from the mechanical model discussed above (Bonazziet al., 2014) is that a positive feedback between growth and polarity can explain the stabilisationduring spore outgrowth. Bonazzi et al. (2014) considered three possible cues that bias the randommotion of the polarisome (polarly localised proteins): curvature of the cell surface, mechanicalstress in the surface, and expansion rate. Only the last cue led to stable cylindrical shapes, whichwas supported by further experiments in which wall expansion was manipulated.Rojas et al. (2011) developed a model coupling the deposition of new material and the me-chanics of the cell wall. They reproduced the morphologies of pollen tubes and were able to12xplain growth oscillations that are observed in rapidly growing tubes. In this model, the rate ofdeposition of wall material decreases with the velocity of cell tip, making a link between exocy-tosis and growth rate. Consequently, the cell may either grow at constant rate or oscillate betweenphases of high deposition and slow elongation and phases of low deposition and fast elongation.This results in either cylindrical or pearled pollen tubes, respectively. At least two hypothesescould account for such negative feedback. A ‘passive’ hypothesis is that when growth velocityincreases, exocytosis becomes relatively too slow to provide materials to the growing tip. An ‘ac-tive’ hypothesis is that a high rate cell wall of expansion may lead to higher membrane tension(due to limited membrane supply), leading to the opening of mechanosensitive channels and theentry of cytosolic calcium that would downregulate the polymerisation of the actin cytoskeletonand thus the delivery of cell wall material (Kroeger et al., 2008; Yan et al., 2009).Rojas et al. (2017) combined experiments in
Bacillus subtilis and a non-spatialised dy-namical model to also propose that enhanced expansion would induce high membrane tension.Mechanosensing would prevent over-expansion of the wall by reducing the supply of wall pre-cursors when membrane tension is too large. It would be interesting to know whether this mech-anisms is involved in regulating cell shape. Overall, the three studies discussed use models tosuggest that the rate of cell wall expansion is sensed, though the mechanisms behind are still tobe identified.
5. Conclusions
The generation of anisotropic shapes in walled cells relies mainly on two strategies. Manycells, such as hyphae, yeasts, root hairs or pollen tubes grow directionally via the supply ofnew material to the cell wall at a precise and restricted location (Tindemans et al., 2006; Drakeand Vavylonis, 2013). Several models for tip growth have been implemented and are able toreproduce most of observed shapes. These computational approaches may give some informa-tion about the mechanics of the cell wall (Dumais et al., 2006; Fayant et al., 2010) or aboutits behaviour with respect to perturbations (Goriely and Tabor, 2008). However, the range ofhypotheses used in these models makes it di ffi cult to know which ones are more relevant to ex-periments. Progress should stem from more quantitative experiments and models, and from thestudy of perturbations in experiments and in models. Rod-shaped bacteria grow di ff usely, re-quiring the mechanical reinforcement of their sides. This reinforcement is likely achieved thanksto the circumferential insertion of glycan strands (Huang et al., 2008; Furchtgott et al., 2011).On a multicellular scale, plants combine those tip growth and anisotropic di ff use growth to ini-tiate organs morphogenesis. Theoretical models indicate that the two mechanisms are necessaryto induce the massive shape changes required for organogenesis. The major actors of this me-chanical control of morphogenesis, each corresponding to one of the strategies discussed here,are the phytohormone auxin, which is involved in the softening of the plant cell wall (Rein-hardt et al., 2000), and cellulose, a sti ff polymer whose oriented deposition leads to sti ff nessanisotropy (Baskin, 2001).Several feedbacks have been identified that may contribute to the maintenance of rod-shapes.Bacteria, thanks to a simple mechanism of curvature-sensing based on the MreB protein, areable to maintain and even to generate de novo cylindrical shapes (Billings et al., 2014; Ursellet al., 2014; Wong et al., 2017). In fission yeast, a precise volume doubling between the germi-nation and the outgrowth is granted by the mechanical rupture of its protective shell (Bonazziet al., 2014). Force-sensing is also involved in organogenesis in the plant shoot apex. Mechan-ical stress, auxin transport, auxin-induced softening and cellulose anisotropy feed back on each13ther in complex loops that may be required for the robustness of organogenesis (Hamant et al.,2008; Heisler et al., 2010). Finally, growth-sensing explains both the random movement and thestabilisation of polarity before and after the triggering of the outgrowth in fission yeast (Bonazziet al., 2014). Growth-sensing might also be relevant for the oscillatory growth in pollen tubes(Rojas et al., 2011). All these studies show how theoretical approaches may help unravelling thecomplex feedbacks that underly organismal growth and robustness of morphogenesis. Simula-tions of these models enable testing alternative hypotheses that can be di ffi cult to di ff erentiateexperimentally or may lead to the identification of key experiments.The molecular actors behind many of these feedbacks are unknown. Curvature-sensing inbacteria could be due to a membrane curvature-dependent binding energy of the protein complexthat includes MreB. Negative curvature and MreB localisation could also be driven by a com-mon signal such as proteins involved in cell wall synthesis (Billings et al., 2014). In fission yeast,mechanisms similar to the oscillatory growth of pollen tubes could explain the stabilisation of po-larity by surface expansion (Yan et al., 2009; Rojas et al., 2011). Alternatively, polarity could bediluted and destabilised in the absence of su ffi cient growth (Layton et al., 2011). Finally, growthcould be involved in the monitoring of cellular dimensions by intracellular gradients (Howard,2012). In the context of the shoot apex, feedback mechanisms are still poorly understood andwe may only speculate. Stretching of the cell membrane or of the cell wall could activate ionchannels or modify the conformation of wall-bound proteins, triggering pathways that impacton microtubules or on PIN1 (Landrein and Hamant, 2013). In the case of auxin transport, analternative hypothesis is the activation of exocytosis and inhibition of endocytosis by the ten-sion in the cell membrane (Hamill and Martinac, 2001). All these hypotheses lack evidence, butnew insights can be expected with progress in cellular and developmental biology, together withphysically-based models of the associated processes.We tried here to highlight concepts and generic mechanisms that hold across kingdoms. Fu-ture directions might stem from enhanced cross-fertilisation between approaches and conceptsdeveloped in the context of specific systems. For instance, detailed mechanical models of thegrowing bacterial cell wall could provide inspiration for the more complex and less organisedcell walls of fungi, oomycetes, and plants. Conversely, continuous models developed for plants,fungi, and oomycetes could be used to test coarse hypotheses in bacterial morphogenesis, beforedealing with more involved molecular details. More generally, modelling morphogenesis at mul-tiple scales, with models assembled as a Russian doll to make links between successive scalesor levels, would allow to deal with a suite of simple models than can be falsified separately andassembled to address more elaborate questions. Acknowledgements
AB is supported by Institut Universitaire de France. AB would like to thank the Isaac NewtonInstitute for Mathematical Sciences for support and hospitality during the programme Growthform and self-organisation when this paper was finalised. This work was partially supported byEPSRC grant number EP / K032208 / References
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