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FL44CH19-Forterre ARI 20 October 2011 16:48 R E V I E W S I N A D V A N C E “Vegetable Dynamicks”: The Role of Water in Plant Movements Jacques Dumais 1 and Yo ¨ el Forterre 2 1 Department of Organismic and Evolutionary Biology, Harvard University, Cambridge, Massachusetts 02138; email: [email protected] 2 IUSTI, CNRS, Aix-Marseille Universit´ e, 13453 Marseille cedex 13, France; email: [email protected] Annu. Rev. Fluid Mech. 2012. 44:453–78 The Annual Review of Fluid Mechanics is online at fluid.annualreviews.org This article’s doi: 10.1146/annurev-fluid-120710-101200 Copyright c 2012 by Annual Reviews. All rights reserved 0066-4189/12/0115-0453$20.00 Keywords plant biomechanics, osmotic pressure, growth, poroelasticity, instability, surface tension, complex fluids Abstract Although they lack muscle, plants have evolved a remarkable range of mecha- nisms to create motion, from the slow growth of shoots to the rapid snapping of carnivorous plants and the explosive rupture of seed pods. Here we review the key fluid mechanics principles used by plants to achieve movements, sum- marizing current knowledge and recent discoveries. We begin with a brief overview of water transport and material properties in plants and emphasize that the poroelastic timescale of water diffusion through soft plant tissue imposes constraints on the possible mechanisms for motion. We then dis- cuss movements that rely only on the transport of water, from irreversible growth to reversible swelling/shrinking due to osmotic or humidity gradi- ents. We next show how plants use mechanical instabilities—snap buckling, cavitation, and fracture—to speed up their movements beyond the limits im- posed by simple hydraulic mechanisms. Finally, we briefly discuss alternative schemes, involving capillarity or complex fluids. 453
Transcript
  • FL44CH19-Forterre ARI 20 October 2011 16:48

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    Vegetable Dynamicks:The Role of Water in PlantMovementsJacques Dumais1 and Yoel Forterre21Department of Organismic and Evolutionary Biology, Harvard University, Cambridge,Massachusetts 02138; email: [email protected], CNRS, Aix-Marseille Universite, 13453 Marseille cedex 13, France;email: [email protected]

    Annu. Rev. Fluid Mech. 2012. 44:45378

    The Annual Review of Fluid Mechanics is online atfluid.annualreviews.org

    This articles doi:10.1146/annurev-fluid-120710-101200

    Copyright c 2012 by Annual Reviews.All rights reserved

    0066-4189/12/0115-0453$20.00

    Keywordsplant biomechanics, osmotic pressure, growth, poroelasticity, instability,surface tension, complex fluids

    AbstractAlthough they lackmuscle, plants have evolved a remarkable range ofmecha-nisms to createmotion, from the slow growth of shoots to the rapid snappingof carnivorous plants and the explosive rupture of seed pods. Here we reviewthe key fluidmechanics principles used by plants to achievemovements, sum-marizing current knowledge and recent discoveries. We begin with a briefoverview of water transport and material properties in plants and emphasizethat the poroelastic timescale of water diffusion through soft plant tissueimposes constraints on the possible mechanisms for motion. We then dis-cuss movements that rely only on the transport of water, from irreversiblegrowth to reversible swelling/shrinking due to osmotic or humidity gradi-ents. We next show how plants use mechanical instabilitiessnap buckling,cavitation, and fractureto speed up their movements beyond the limits im-posed by simple hydraulic mechanisms. Finally, we briefly discuss alternativeschemes, involving capillarity or complex fluids.

    453

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    1. INTRODUCTIONPlants offer some of the most elegant applications of fluid mechanics principles found in nature.Trees are little more than hydraulic systems that take water deep into the ground and elevate itto the leaves. The branching of the shoots and roots, the anastomosing venation of leaves, andthe structural organization of plant tissues all speak to their transport function. Given this, it isnot surprising that plants have attracted attention in applied fluid mechanics (Canny 1977, Rand1983). The first detailed study of the movement of water within plants is provided by Haless(1727) Vegetable Staticks (Figure 1a). Hales (1727) not only described with great accuracy thetranspiration stream of plants, he also sought to interpret his observations in light of the fluidmechanics of his time: The sap vessels are so curiously adapted by their exceeding fineness, toraise the sap to great heights, in a reciprocal proportion to their very minute diameters.

    Since the pioneering work of the Darwins (Darwin 1875, Darwin & Darwin 1880)(Figure 1b), the question of how plants move in the absence of muscle has attracted the interestof many scientists ( Jost & Gibson 1907, Ruhland 1959, Hart 1990). From a biological perspec-tive, the physiology of plant movements is central to our understanding of plant developmentand plants responses to environmental stimuli such as light and gravity (Gilroy & Masson 2008,Moulia & Fournier 2009). In engineering and applied sciences, these nonmuscular movementshave provided inspiration for biomimetic design in the area of microfluidics and robotics (Taya2003, Burgert & Fratzl 2009, Martone et al. 2010).

    The goal of this review is to present some key fluid mechanics principles used by plants toachieve movement, highlighting recent work performed at the frontier of mechanics and biology.We do not address processes involving only water transport without organ motion, as seen, forexample, in the ascent of sap or the translocation of sugars in vascular systems. These topicsconstitute a broad field of research and have already been the subject of many monographs andreviews (Tyree & Zimmermann 2002, Holbrook & Zwieniecki 2005), including reviews in thisseries (Canny 1977, Rand 1983).

    Still the diversity of mechanisms that fit within the scope of this review is vast. From the slowgrowth of shoots and roots to the opening and closing of the minute stomata at the leaf surface,plants exhibit movements on a wide range of length scales. Most of these movements are slow,but some compete in speed with those encountered in the animal kingdom and are used by plantsto trap prey or disperse their seeds (Sibaoka 1969, Hill & Findlay 1981).

    a b

    Figure 1Vegetable staticks and dynamicks. (a) Experimental setup used by Hales (1727) to collect the watertranspired by various plants. (b) A recording of leaf movement by Darwin & Darwin (1880).

    454 Dumais Forterre

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    We begin with a brief overview of the plant cell and tissue structure and a discussion of thevarious mechanical and hydraulic processes (e.g., osmosis, elasticity) enabling plants to move(Section 2). Particular attention is given to how these processes set the timescales of the variousmovements. In Section 3, we discuss hydraulic movements at the cell and tissue level, fromirreversible growth to reversible swelling/shrinking. In both cases, the driving force for motionis the high differential water pressure supported by the plant cell wall. In Section 4, we show howmechanical instabilitiessnap buckling, cavitation, and fracturecoupled with water transportallow plants to overcome the poroelastic limit set by the diffusion of water through tissue. Finally,in Section 5, we briefly discuss alternative strategies for movement broadly defined involvingcapillarity and complex fluids.

    2. FUNDAMENTALSA bouquet of tulips left standing without water will gradually droop, the flaccid stems unable tosupport the weight of the blossoms. If water is provided again, the stems will soon straighten, andthe bouquet will regain its former glory. Although the wilting of a flower may seem mundane,it encapsulates the essence of the mechanism used by plants to achieve movement. To move,plants drive water in and out of their cells by manipulating the osmotic gradients across theirsemipermeable membranes. The local changes in cellular volume and tissue stiffness enable thelarge-scale tissue deformations required for motion.

    Therefore, plant motion is fundamentally a mechanical problem whose key features are rootedin the structure and physiology of plant cells. Within the context of mechanics, the plant bodycan be decomposed into two simple phases: a fluid phase representing as much as 75% of the totalmass of fresh tissue and a solid phase made largely of the cellulosic walls that surround every plantcell. At this level, plants do not differ significantly from algae, fungi, and other organisms withwalled cells. We therefore have sampled freely between these groups to emphasize the generalityof the principles at work, but also to highlight the great ingenuity with which nature has soughtto endow these organisms with the power of movement.

    2.1. Plant Cell and Tissue StructureA fundamental difference between plants and animals is that plant cells are surrounded by a thinbut stiff cell wall made of highly organized cellulose microfibrils embedded in a pectin matrix(Preston 1974, Taiz & Zeiger 2002, Baskin 2005) (Figure 2b). This structural difference pre-vents plant cells from using soft contractile proteins (such as the actomyosin system of musclefibers) to deform and generate movement. However, the stiff wall allows plant cells to sustain alarge internal hydrostatic pressure known as turgor. A turgor pressure of approximately 0.5 MPa(5 bars) is common (Green et al. 1971, Zhu&Boyer 1992) (Figure 2a). Cells develop this pressureby maintaining an osmotic gradient between their cytoplasm and the environment, thus allowingwater to move into the cell and put the wall under tension. As shown below, this high turgorpressure provides the force for changing the cell volume and is thus the main motor for growthand motion in plants.

    With regard to plant tissues or entire organs, water can be found in two separate but inter-penetrating compartments: the symplast and the apoplast (Figure 2b). The symplast is defined asthe volume contained within the plasma membrane of cells and thus under direct osmotic controlby the cells. The apoplast is the dual of the symplast and includes all the volume taken by cellwalls and intercellular spaces. Direct symplastic flow between cells is possible through intercellular

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    Fern leptosporangium

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    Figure 2Water in plants. (a) The range of water pressures found in plants as compared with other familiar systems.Plants are the only living organisms that use water in a state of high tension, as much as 300 bars in the caseof the fern leptosporangium. (b) The principal paths along which water flows. The cell-to-cell path!comprises both the transmembrane path, where water moves across the semipermeable plasma membrane,and the direct symplastic path through the plasmodesmata that does not cross the plasma membrane (doubleblue line). In the apoplast path", water moves through the relatively porous cell walls and intercellularspaces without crossing any membranes. Finally, evaporative loss at the plant surface# plays an importantrole in water relations and also drives many plant movements.

    Water potential: thechemical potential ofwater per unit volumerelative to a referencestate

    bridges known as plasmodesmata. Conversely, the porous nature of the cell walls that constitutethe apoplastic space provides an alternative path of high conductivity for water flow.

    2.2. Water: The Prime MoverThe flow of water in living plant tissues shares many similarities with flow in porous media, withthe notable exception that osmotic gradients must be considered alongside the pressure gradients.Depending on the length scale at which flow is considered, the permeability associated with thepressure and osmotic gradients will vary slightly. Therefore, we consider in turn water flow at thecellular level and at the tissue level.

    2.2.1. Water transport across the cell membrane. At the cellular level, the water flux j (inmeters per second) across a perfect semipermeable membrane, and thus the rate of change in thecell volume V, is driven by the change in the chemical potential of water, according to the relation

    dVdt

    = j A= ALp!" = ALp (!P +!pi ), (1)where A is the cell membrane area; Lp is the hydraulic conductivity of the cell membrane (inmeters per second per pascal); " = P pi is the water potential (in pascals); P is the hydrostaticpressure minus the atmospheric pressure; pi = cRT is the osmotic component, with c the soluteconcentration, R = 8.32 J mol1 K1 the gas constant, and T the temperature (in Kelvin); and! stands for the difference between the inside and outside of the cell (Dainty 1976, Finkelstein1987, Kramer & Boyer 1995, Nobel 1999). In the field of animal physiology, Equation 1 is known

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    as the Starling equation and is used to quantify the rate of plasma filtration and resorption incapillary beds. In the context of plant cells, Equation 1 states that changes in the volume of cellsare driven by modifications of the turgor pressure or solute concentration from their equilibriumvalues. The rate of water flow depends on the membrane conductivity Lp, which is a crucialquantity for water relations in plants. Values of the membrane conductivity span a wide range,1013 Lp 1011 m s1 Pa1 (Steudle 1989), and may be regulated by the opening or closingof water pores, called aquaporins, which are under tight physiological and molecular regulation(Maurel et al. 2008).

    2.2.2. Water transport across plant tissue. The previous water relations at the cellular levelapply to the description of flow in tissues at the macroscopic level. For instance, the assumptionof only cell-to-cell transport across the cell membranes (path 1 in Figure 2b) yields a Darcy-likerelation between the water flux and the water-potential gradient, with a Darcy permeability (insquared meters) Lp R, where (in pascal seconds) is the water viscosity (Philip 1958). How-ever, the situation in most tissues is more complex because water may also flow in the cell wallcontinuum (apoplast pathway), whose permeability kwall depends on both the cell wall volumefraction in the tissue and the cell wall permeability kwall (Figure 2b). In many cases (roots andstems), both pathways are of similar conductivity, and one may define a single effective permeabil-ity k encompassing both the cell-to-cell and apoplast pathways, with typical values 1020 k 1019 m2 (Molz & Ikenberry 1974, Molz & Ferrier 1982, Steudle 1992). However, although os-motic and turgor gradients may drive water flow across the cell membrane, only pressure gradientslead to significant bulk flow in the apoplast because the cell wall continuum is permeable to mostsolutes (Steudle 1989, 1992). Thus both driving forces are not equivalent at the tissue level.

    2.3. Material Properties of the Plant Cell WallThe plant cell wall is a living material and as such must be approached mechanically with somecaution. For example, whatmay look like simple viscosity at themacroscopic level can in fact reflectthe addition of mass and a chemically mediated remodeling of wall material at the microscopiclevel. The constitutive modeling of materials that add mass and remodel as they deform is anactive area of research (Ambrosi et al. 2011). Here we focus mostly on the macroscopic propertiesof cell walls, as they suffice in explaining plant movements.

    2.3.1. Elastic regime. The plant cell wall behaves elastically for a narrow range of strains (5%or less), with the exception of a few specialized cells, such as the guard cells of the stomata (seeSection 3.2), in which higher wall strains can also be reversible. The Youngs modulus Ewall of thecell wall of actively expanding cells is typically less than 1 GPa (Probine & Preston 1962), whereasfor wood fibers, it exceeds 25 GPa (Gibson & Ashby 1999). An alternative measure of elasticityis the bulk modulus of the cell = V (dP/dV ) (Figure 3a). The bulk modulus characterizes howchanges in volume are related to changes in turgor pressure, and has a typical value between 1 and50 MPa (Steudle et al. 1977, Cosgrove 1988). Although it is defined in the same way as the bulkmodulus of standard engineering materials, the interpretation of the cellular bulk modulus can bechallenging as it reflects not only the elasticity of the cell wall, but also the geometry of the cell(Wu et al. 1985, Cosgrove 1988). The cellular bulk modulus is nonetheless a useful parameter asit plays a pivotal role in setting the timescale of cellular responses to changes in osmotic balance.

    At the tissue level, the relation between the macroscopic Youngs modulus of the tissue andthe cell properties (wall elasticity, cell size) can be approached from the standpoint of cellu-lar materials (Gibson & Ashby 1999, Gibson et al. 2010). As long as the turgor pressure ishigh and the cell walls stretched, the tissue Youngs modulus scales with the cell bulk modulus,

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    yoelInserted Textand R the typical cell size.

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    0.600.550.500.45

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    Figure 3(a) Elastic and (b,c) plastic behavior of the cell wall as observed in Chara and Nitella internodal cells. (a) The pressure-volume curve isshown for a single internodal cell in the elastic regime (Kamiya et al. 1963). Note the hysteretic response. (b) Increases or decreases inturgor pressure lead to an abrupt (elastic) change in cell length followed by a gradual transition to a new elongation rate (Proseus et al.2000). (c) The elongation rate depends strongly on the cells pressure and shows a marked yield pressure (approximately 50% of thenormal pressure) below which growth is not possible (Proseus et al. 2000).

    Tropic response:movement determinedby the direction of thestimulus (gravity, light,touch); examplesinclude phototropism,gravitropism, andthigmotropism

    Nastic response:movement that occursin a directionindependent of thestimulus (e.g., thefolding of the Mimosapudica leaf and closingof the Venus flytrap)

    E 10 MPa. However, E decreases sharply when the turgor pressure drops to zero becausethe cell walls are no longer stretched and can bend easily (Nilsson et al. 1958,Warner et al. 2000).

    2.3.2. Plastic regime and growth. Historically, the constitutive behavior of the growing cell wallhas been described as a Bingham fluid (Lockhart 1965, Green et al. 1971, Cosgrove 1985, Proseuset al. 1999). Bingham fluids differ from ordinary viscous fluids in that they deform irreversiblyonly for stresses that exceed a plastic yield stress y. For uniaxial deformation, the rate of plasticdeformation is written as ( = ( y ) for > y , where is the material extensibility (aninverse viscosity). Lockhart (1965) was the first to apply this constitutive law to wall extension andgrowth in plants, which was later extended to incorporate a viscoelastic behavior (Ortega 1985,1990). Extensions of the simple uniaxial model to anisotropic materials under multiaxial stress arenow available (Dumais et al. 2006, Dyson & Jensen 2010).

    The simple constitutive law adopted byLockhart is well supported experimentally. A number ofinvestigators have demonstrated that the relative rate of cell expansion is not simply proportionalto the cell pressure (Green et al. 1971, Proseus et al. 1999). For example, small-step increases ordecreases in pressure (

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    Figure 4Swelling/shrinking timescales in plant cells and tissues. (a) Single-cell relaxation in a giant algal cell (Chara corallina internode)measured by a cell pressure probe (Steudle 1993) (data from Ye et al. 2006). A pressure step is applied by suddenly injecting a smallvolume of water in the cell. In this system, V/A = 200 m and = 30 MPa. The cell relaxation time is cell = 3.5 s, which gives acell membrane conductivity Lp 2 1012 m s1 Pa1. (b) Reversible swelling and shrinking of a plant tissue (pealed pea epicotyl 1 cmin length) suddenly immersed in baths of different solute concentrations (upper panel ) or in a fresh solution after desiccation in air(lower panel ). The measured tissue relaxation time (poroelastic time) is p = 4080 s. In this radial water transport geometry,p = 0.022 L2/D (Philip 1958), where L = 1.6 mm is the segment diameter, which gives a coefficient of diffusion D 109 m2 s1.Data taken from Cosgrove & Steudle (1981).

    Braam2005), with no clear reference to the underlying physicalmechanism.A key suggestionmaderecently by Skotheim & Mahadevan (2005) is that the timescale for water transport constrainsthe maximal speed of hydraulic movements in these soft nonmuscular systems, thus providinga physical basis for the classification of motion in plants and other walled organisms such asfungi.

    2.4.1. Cell relaxation time. We first address this question at the cellular level. A cell perturbedfrom equilibrium by a small sudden change in the osmotic potential or turgor pressure relaxesexponentially with a timescale given by

    cell = VA( +!pi )Lp RLp

    , (2)

    where R = V/A is the typical cell radius, and $ !pi , as is the case for most turgid cells(Dainty 1976). Typical measurements in giant algal cells of R = 200 m, = 30 MPa, andLp = 2 1012 m s1 Pa1 give cell = 3.5 s (Figure 4a).

    The cell relaxation time (Equation 2)may be interpreted as the shortest response time of a plantcell to small water-potential perturbations and thus provides a bound for hydraulic movementsat the cellular level. Figure 5a plots the timescale of motion as a function of the typical cellsize R for a wide range of unicellular movements in plants and fungi, together with the boundarygiven by Equation 2. Systems with > cell can rely on water transport to swell or shrink, whereassystems with < cell must use other mechanisms, as shown in Sections 4 and 5. The dependenceof cell on the cell size R shows that water transport may be fast if a cell is sufficiently small, as inthe case of the rapid swelling of nematode-trapping fungi (see Section 3.1).

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    Ballistospores(surface tension)

    NitellaPollen tubeSporangium (opening)OedogoniumStomata Chara (relaxation)Arthrobotrys brochopaga (fungus)Zoophagus insidians (fungus)A. dactyloides (fungus)Dactylaria brochopaga (fungus)Utricularia (trapdoor)PilobolusSporangium (closing)Ballistospore

    Lonicera japonicaL. sempervirensPinguicula vulgarisIpomoea nilPharbitis nilArabidopsis thalianaOryza sativaDesmodium gyransMimosa pudicaDionaea Stylidium crossocephalumS. graminifoliumS. piliferumUtricularia (trapdoor)AldrovandaCatasetumLepidium campestreRuellia brittonianaEcballium elateriumHura crepitansArceuthobium

    Figure 5Classification of (a) unicellular and (b) multicellular movements in plants and fungi, based on the timescale of water transport (from anoriginal idea of Skotheim &Mahadevan 2005). The plots give the duration of movement as function of (a) the cell radius R and (b) thetissue size L, defined as the smallest macroscopic moving part. The order of the labels in the figure key coincides with their order in thefigure from top to bottom. In panel a, the solid blue line gives the cell relaxation time (the fastest cell swelling) cell = R/(Lp ), with = 30 MPa and Lp = 2 1012 m s1 Pa1, and the dashed blue line gives the Lockhart time (the fastest cell growth not limited bywater transport) Lockhart = R/(!piLp ), with !pi = 0.5 MPa. In panel b, the solid blue line gives the poroelastic time (the fastest tissueswelling) p = 0.022 (L2/D), with D = 109 m2 s1. The solid red line gives the timescale for elastic wave propagation (the fastestelastic motion) el = L/E, with = 1,000 kg m3 and E = 10 MPa.

    2.4.2. Poroelastic time. The previous relation (Equation 2) holds at the cellular level. However,the swelling or shrinking of a tissue requires not only a local change in cell volumes, but also thetransport of water from one part of the tissue to another. The flow of water in a soft porous mediasuch as plant tissue is a diffusive process, whose timescale over a tissue of size L, known as theporoelastic time p, is given by

    p L2

    D L2

    kE, (3)

    where D = kE/ is a diffusive coefficient that depends on the Darcy permeability of the poroustissue k, the (undrained) Youngs modulus of the tissue E, and the water viscosity (Biot 1941,Wang 2000). Philip (1958) was the first to derive a diffusion-like equation for water transportthrough plant tissues assuming cell-to-cell transport only. This model was further extended toincorporate both the cell-to-cell and the apoplast pathway (Molz & Ikenberry 1974, Molz &Ferrier 1982, Steudle 1992). Typical values for plant tissues (k = 1020/1019 m2, E = 10 MPa,

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    and = 103 Pa s) give D = 1010/109 m2, compatible with swelling experiments of tissues(Figure 4b) (Molz & Boyer 1978, Cosgrove & Steudle 1981, Steudle 1989).

    The poroelastic time p (Equation 3) may be interpreted as the fastest possible water-drivenmovement at the tissue and organ levels (Skotheim&Mahadevan 2005). Its strong size dependence(L2) shows that hydraulicmovements are increasingly less efficient in terms of speed as the systemsize increases. Figure 5b presents the duration of plant movement as a function of the typicaldistance L through which the fluid is transported for a wide range of movements and timescales.The boundary = p naturally separates two categories of movements dominated either byswelling/growth or by mechanical instabilities.

    In the following, we present some specific examples of plant movement mechanisms in lightof this classification. We first address slow hydraulic movements (irreversible growth, reversibleswelling/shrinking). We then shift to rapid movements that cross the poroelastic boundary, inparticular those using a mechanical instability to amplify their speed.

    3. HYDRAULIC MOVEMENTS IN PLANTS

    3.1. Growth and Growth Movements

    The most ubiquitous but also least obvious movements in plants are associated with growth. Ittook the patient eye of Charles Darwin (Darwin & Darwin 1880) to finally draw greater attentionto them. The elongation of stems is a slow but vital race toward the sun. Plants left behind arebound to spend their lives in the shadow of their taller neighbors. Given the selective pressure forrapid growth in many environments, one may ask what limits the rate of plant growth. Lockhart(1965) provided the first model to address this question at the cellular level. For a cell to expand,two processes must occur concomitantly: The cell wall must increase its surface area, and watermust enter the cell to increase its volume. Lockharts model for a cylindrical cell of radius R andwall thickness h predicts the following governing equation for the relative rate of volume increase:

    ( = 1V

    dVdt

    = Lp (R!pi hy )hLp + R2 . (4)

    If typical values for cell geometry, hydraulic conductivity, and wall extensibility are considered, itis seen that hLp $ R2. Lockharts equation thus simplifies to

    (l imited = (R!pih

    y). (5)

    This limit is known as the extensibility-limited regime for cell expansion. In other words, thecharacteristic strain rate of growing plant cells is set by how fast the cell can extend its wall and notby its ability to take up water through the plasmamembrane. Given that the extensibility is set bythe rate at which the wall is synthesized and assembled by the cell, the extensibility-limited regimeis really a statement about the cells maximal metabolic rate. In that context, it is a remarkableobservation that some cells can achieve local strain rates as high as 0.25 min1 (Rojas et al. 2011)(an equivalent wall viscosity 1/ 10 GPa s, assuming typical values R/h 100 and !pi 0.5 MPa).

    An elegant example of growth-mediated movement is seen in the phototropic response of thePhycomyces sporangiophore, a cell of gigantic size (as much as 10 cm) that raises the sporangiumand its spores above the boundary layer for dispersal (Figure 6a). The zone of elongation isconfined to a 3-mm region just below the terminal sporangium. Within this region, strain ratescan reach values as high as 0.05 min1. The high strain rates and relatively narrow sporangiophore

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    1 mm

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    Figure 6Growth movements at the cellular and tissue scales. (a) Phototropic bending of the Phycomycessporangiophore in response to unilateral light coming from the right. Images were taken at 5-min intervals(Shropshire 1963). (b) Bending movement induced by differential growth of a cyclamen flower stalk(MacDonald et al. 1987). The numbers indicate the times (in hours) at which the pictures were taken. Thecoiling proceeds over several days until the terminal seed pod is buried into the ground.

    stalk allow for rapid bending (37 min1) in response to unidirectional light (Figure 6a) (Castle1962). Similar bending responses, albeit slower, are also seen in stems and roots (Figure 6b).

    In some special cases, cells can increase in volume at a relative rate that is nearly equal tothe maximum rate the membrane conductivity will allow (Equation 1). This regime is known asconductivity-limited growth (hLp ' R2). In this limit, the Lockhart equation takes the form

    (Lplimited = Lp(R!pi hy )

    R2. (6)

    Figure 5a shows the Lockhart timescale, defined as

    Lockhart = RLp!pi 1

    (Lplimited. (7)

    The Lockhart timescale may be interpreted as the fastest possible cell growth not limited by waterdiffusion across the membrane. It is usually much larger than the cell relaxation time ( $ !pi ),except in cells in which the turgor pressure is very high and the wall is soft like stomata.

    One way cells reach the conductivity limit (Equations 67) is by assembling the wall surfacenecessary to increase cell volumebefore expansion.The algal cellOedogonium offers a clear example(Figure 7a). Cell expansion in these algae is coupled to cell division and begins with the formationof an internal flap of wall material at one end of the cell. Because of its position within the externalload-bearing wall, this flap can develop without having to support the turgor pressure of the cell.When the cell is ready to divide and expand, a fracture develops in the outer wall, thus transferringthe load to the unstretched wall flap. Consequently, the initial phase of expansion is mainly limitedby the rate at which water can enter the cell. Actively growing cultures ofOedogonium are describedas twitching, providing some indication of the bursting nature of cell expansion following therelease of the old wall. One observes in Figure 5a that Oedogoniums growth rate is indeed closeto the Lockhart limit.

    A similar mechanism has been put to good use in nematode-trapping fungi. These soil fungihave evolved a simple noose that they use to capture the nematodes with whom they share theirenvironment (Figure 7c). Although the capture mechanism has not been fully elucidated, it isbelieved that a load-bearing wall layer loses its mechanical integrity upon stimulation of the noosecells by a passing worm. The ensuing sharp decline in turgor pressure puts the cell out of osmotic

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    a b

    c

    t = 0 s t = 0.12 s t = 0.18 s

    Growthscar

    Slit

    Wallring

    5 m

    10 m

    10 m

    d

    Figure 7Conductivity-limited growth. (a) Expanding inner wall ring in Oedogonium (Pickett-Heaps 1975). (b) Wall-ring development and early phase of expansion (Pickett-Heaps 1975). The wall ring develops inside the cellat the level of a slit in the load-bearing wall. Fracture of the outer wall along the slit puts the inner wall ringunder tension, allowing for rapid cell elongation. (c) Constricting ring of the nematode-trapping fungusArthrobotrys brochopaga. (d ) A trapped nematode. Panels c and d taken from Nordbring-Hertz et al. (2006).

    equilibrium, thus allowing water to rush in. The rapid swelling of the cells closes the noose, leavingthe snared prey to be slowly invaded by fungal hyphae and digested.

    For cells bathed in water, the conductivity of the cell membrane is such that water uptakerarely limits growth. However, in multicellular tissues, cells have access to water only throughneighboring cells.Onemay askwhether, in that context, water flowmaynot limit the rate of growth(Molz & Boyer 1978, Boyer & Silk 2004). Simple scaling arguments may be used to address thisquestion. For uniform growth to occur in a plant tissue, this tissue must maintain a constant waterflux given by the Darcy law (k/)!P/L, where L is the typical size of the growing organ and!P is the pressure difference. This flux is related to the growth rate ( by volume conservation(k/)!P/L L(. One may assume that the process is not conductivity limited as long as thepressure drop is small compared with the typical cell turgor P, yielding ( ' kP/L2. The inverseof this maximal growth rate sets the tissue Lockhart timescale:

    Lockhart tissue L2

    kP. (8)

    This timescale is the analog at the tissue level of the Lockhart timescale for the cell (Equation 7).It may be interpreted as the fastest possible uniform tissue growth not limited by water transport.For a while, there has been quite a debate in the plant physiology community on whether thesegrowth-induced gradients of pressure could play an important role in plant growth (Steudle 1989).For rapidly growing tissues (coleoptile, roots), they might not be negligible and may play a rolein the regulation of growth movements (gravitropism, phototropism, thigmotropism).

    3.2. Actuation Driven by Osmotic GradientsThe second main class of water-driven movements in plants is associated with small reversiblechanges in the cell volume within the elastic range of cell wall deformation. These modifications

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    Figure 8Actuation in plants driven by osmotic gradients. (a) Relationship between the stomata aperture and theturgor pressure inside the guard cells, measured by confocal microscopy. Figure adapted from Franks et al.(2001). The left panel shows the stomata on the surface of a broad-bean leaf (Vicia faba). (b) The slowrecovery of the main pulvinus ofMimosa pudica imaged by nuclear magnetic resonance, showing adisplacement of water from the upper to the lower half. The plot gives the distribution of water as functionof time, together with the petiole to stem angle. Figure adapted from Tamiya et al. (1988).

    may be driven by an active transport of solutes (ions) across the cell membrane by specializedpumps. Although the variations in volume involved are usually small, the resulting change in turgorpressure may be very large as the cell wall is stiff (large bulk modulus). The additional couplingwith the geometry and structural properties of the cell, tissue, or organ allows the amplifyingmechanism to convert small reversible changes in volume into large movements.

    The opening and closing of stomata are probably among the most important and thereforebest-studied reversible movements in plants (Meidner & Mansfield 1968) (Figure 8a). Stomataare small pores on the surface of the leaves that control the leaf transpiration and gas exchangewith the atmosphere, on a timescale ranging from a day (diurnal rhythm) to a few minutes (short-term response to environmental change) (Woods & Turner 1971, Buckley 2005, Kaiser & Grams2006). In most plants, the stomatal complex consists of two kidney-shaped guard cells that flanka central pore (Taiz & Zeiger 2002). When the turgor pressure inside the guard cells increasesowing to accumulation of solute (up to P 5 MPa), the cells swell reversibly (by 20%40% involume) and become more curved in shape, pushing apart the surrounding cells and opening thepore (Figure 8a). When the turgor pressure decreases, the two guard cells are pressed together

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    by the surrounding cells, and the pore closes. The relationship between the stomatal aperture andthe turgor pressure of the guard cells and the surrounding epidermal cells has been studied bothexperimentally and theoretically (Aylor et al. 1973, de Michelle et al. 1973, Cooke et al. 1976,Franks et al. 2001, Franks 2003, Buckley 2005) (Figure 8a). The bending of the guard cells asthey swell comes from the large asymmetry in wall thickness and the mechanical anisotropy of thecellulosic network.

    Other classic examples of actuation in plants are the circadian and light-induced leafmovementsof several plants due to a specialized multicellular motor organ, called the pulvinus, located at thebase of the leaf petiole (i.e., where the leaf comes in contact with the stem). These movementsare usually under slow diurnal rhythms but can be as fast as seconds in the case of the leaves ofthe sensitive plant Mimosa pudica. The current paradigm attributes all pulvinus movements to adifferential osmotic swelling/shrinking on opposite sides of the pulvinus, whose location at thebase of the leaf acts as a lever to amplify the leaf s angular motion (Hill & Findlay 1981, Moran1990) (Figure 8b). The rapid closing of M. pudica seems to have attained the fastest possiblehydraulic motion (Figure 5b) according to the size of its pulvinus [pulvinus diameter L 200 m(Weintraub 1952), poroelastic time b 1 s]. However, to our knowledge, no direct evidence forrapid water flow in this system exists.

    3.3. Passive Actuation Driven by Humidity GradientsOsmotic gradients are not the only mechanisms for water exchange between cells and their sur-roundings. When cells are exposed to a dry atmosphere with a low partial water pressure, evap-oration causes the volume to change as well. Many passive motions in plants are driven by thesehumidity (water-potential) gradients between the cell and the ambient air. The water potentialof air is a function of the water-vapor partial pressure Pvap through the classical vant Hoff rela-tion "vap = (RT /Vw) ln[Pvap/Psat(T )], where Vw 18 cm3 mol1 is the partial molar volumeof liquid water and Psat(T ) the saturation water pressure (Atkins & de Paula 2002). For a rela-tive humidity 100 Pvap/Psat(T ) = 50%, the water potential in air is very low (approximately94 MPa) and, together with water cohesion properties, may lead to large negative pressure incells with stiff walls (see Section 4.2) and force other cells to crumple tightly to accommodate thelost volume.

    Pollen grains are routinely exposed to harsh osmotic environments and have therefore evolveda coping mechanism. The surface of pollen grains comprises two regions: the porous aperturesand the water-impermeable interapertural areas (Figure 9a,b). The permeability of the aperturesis necessary to allow communication and exchange of water with the receptive surfaces of flowers.The same permeability, however, threatens the survival of theminute pollen grains while in transitfrom one flower to another. The solution found by pollen grains is to design their apertures suchthat they fold inwardly when the pollen loses water, thus effectively preventing further evaporation(Figure 9a,b) (Katifori et al. 2010). Unlike crumpling, the regular folding of pollen grains is fullyreversible when water becomes available again.

    3.4. Hygroscopic MovementsHumidity-drivenmovements also occur in dead cells at the tissue level (sclerenchymal tissue) ( Jost&Gibson 1907). Sclerenchymal tissue typically consists of fiber cells with walls comprising severallayers of oriented cellulose fibrils. When absorbing/expelling water in response to changes in airhumidity, the tissue expands/shrinks anisotropically, perpendicular to the fibrils orientation (Fahn&Werker 1972, Burgert & Fratzl 2009). Asymmetry in the orientation of the fibrils at the organ

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    Num

    ber o

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    t (min)t (s)0 400 800 1,200

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    Figure 9Passive movements driven by changes in humidity in (a,b) living cells and (cf ) dead tissues. The folding(self-sealing) response of (a) lily and (b) Euphorbia pollen grains. (c,e) The drilling motion of the awn ofErodium (Evangelista et al. 2011). (e) Number of turns of the awn as a function of time for wetting and dryingtransients. The wetting time constant is = 330 s, and the drying time constant is = 400 s. (d,f ) Theopening motion of conifer cones, showing much longer time constants in accordance with the larger system(Reyssat & Mahadevan 2009).

    level then converts this local swelling/shrinking to a global bending movement, which can drive,for example, the opening and closing of a pine cone (Dawson et al. 1997, Reyssat & Mahadevan2009) (Figure 9d, f ), the penetration of seeds into soil (Elbaum et al. 2007, Evangelista et al. 2011)(Figure 9c,e), and the opening of seed pods (Armon et al. 2011). The difference between theseactuation mechanisms and those seen previously with living cells is that, in dead tissues, there areno longer cell membranes to maintain an osmotic gradient. Still, in some cases, a turgor gradientmay bemaintained at equilibriumwith the atmosphere with the cell wall acting as a semipermeablemembrane between the vapor and liquid phase (Wheeler & Stroock 2008). We note that a relatedmechanism of anisotropic swelling/shrinking driven by the cell wall architecture applies in thecontrol of branch movements in woody organs (reaction wood), although on a different timescaleand length scale (Niklas 1992).

    4. BEYONDWATER DIFFUSION: INSTABILITYAND FLUID-SOLID COUPLINGThe hydraulic movements discussed above are constrained in terms of speed by the timescale ofwater diffusion p = L2/kE (poroelastic time), especially so as the system size increases (see

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    Section 2). Yet, from carnivorous plants to seed dispersal, many plant motions cross this hydraulicboundary to generate some of the fastest motions ever recorded in living systems (Figure 5).The strategy to reach these speeds is based on a simple principle: mechanical instability. A waterflowdriven either by osmotic or humidity gradientsfirst slowly stores elastic energy in the cellwalls, which is prevented from being released by some energy barrier. Above a critical threshold,the barrier is overcome, and the elastic stress is suddenly released. In this section, we discuss thesechampions of the plant kingdom, in light of recent discoveries made possible by the use of high-speed video.Wenote that the speedof these elasticmovements is expected to be limited, ultimately,by the speed of elastic waves (Skotheim & Mahadevan 2005). The timescale el = L/E forelastic waves to travel a distance L, where is the tissue density, is given in Figure 5b. No plantmovements are found below this limit.

    4.1. Snap-Buckling InstabilitiesGeometric frustration is among the most elegant ways for a plant to accumulate elastic energy topower motion. It is based on the idea that a thin shell, under a specific mode of loading, can belocked in a configuration that represents a local energy minimum rather than the global energyminimum. The transition between the two configurations often involves extensive stretching ofthe shella costly mode of deformation in term of elastic energy (Landau & Lifshitz 1986).However, the accumulation of elastic energy with increasing load can allow the shell to cross theenergy barrier. It then snaps into the configuration of lowest energy, releasing the stored elasticenergy. In this section, we give two examples of rapid movements in carnivorous plants that use asnap-buckling instability to catch prey.

    4.1.1. Venus flytrap. The rapid closure of the Venus flytrap (Dionaea muscipula) is one of thebest-known rapid motions in the plant kingdom and led Darwin (1875) to describe the plant asone of the most wonderful in the world (Lloyd 1942, Juniper et al. 1989) (Figure 10a). Closureof the trap is initiated by the mechanical stimulation of one of the trigger hairs [usually twicewithin 20 s (Brown 1916)], which elicits an electrical action potential that spreads over the leaf inless than 1 s (Burdon-Sanderson 1882, Stuhlman&Darder 1950, Sibaoka 1969, Hodick & Sievers1989, Volkov et al. 2007). The trap then shuts in a few tenths of second, which seems too fast tobe accounted for by pure water transport across the 0.5-mm leaf thickness (Figure 5b).

    A recent study suggests that the mechanism of closure of the Venus flytrap involves a snap-buckling instability, analog to the buckling of an elastic shell (Forterre et al. 2005). The two lobesof the trap are curved outward in the open state and curved inward in the closed state (Figure 10b).Upon triggering, the lobes actively change their natural curvature in the direction perpendicularto the midrib. However, because of the geometric constraint of the doubly curved lobe, this activebending causes the trap to accumulate stretching energy, until the stored elastic energy becomesso large that the trap snaps shut rapidly (Figure 10c). Although the lobes behave like curved shells,the closing dynamics is much slower and damped than what one would expect for a purely inertialsnapping of an elastic shell. It has been proposed that, within the hydrated soft porous tissueof the leaf, passive flow induced by bending provides viscous resistance that balances the elasticenergy, in agreement with a poroelastic shell dynamics model (Figure 10c) (Forterre et al. 2005).Biomimics of this mechanism have been applied in the context of microfluidics (soft actuators)(Kim & Beebe 2007), robotics (Lee et al. 2010), and smart surfaces (Holmes & Crosby 2007).

    Whereas the role of elastic instability in the trapping mechanism seems to be supported ata macroscopic level, the mechanisms by which the plant actively changes its natural curvatureand bend are still a matter of debate (von Guttenberg 1925, Lloyd 1942, Hill & Findlay 1981,

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    0.1 0.1Local mean

    curvature m (mm1)

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    Figure 10Snap buckling in carnivorous plants: (ac) Venus flytrap and (df ) bladderworts. (a) Venus flytrap in the (right panel ) open and (leftpanel ) closed state. (b) Dynamic sequence of the leaf closure measured using three-dimensional reconstruction. (c) The spatiallyaveraged mean curvature m (blue) and the spatially averaged Gaussian curvature g (red ) as a function of time (the trap was triggered att = 0). The solid line corresponds to the poroelastic theoretical model. (d ) Sketch of a bladderworts bladder (side view) and top viewof Utricularia inflata in the deflated ready-to-catch state and just after triggering. (e) Inversion of the trapdoor and buckling of themedian door axis visualized by light sheet fluorescence microscopy. ( f ) Numerical simulation of the trapdoor opening. Panels ac takenfrom Forterre et al. (2005) and panels df from Vincent et al. (2011).

    Cell pressure probe:measures the turgorpressure in single plantcells by means of amicrocapillaryconnected to apressure gauge

    Williams & Bennett 1982, Hodick & Sievers 1989, Volkov et al. 2008). Direct measurements ofthe cell turgor pressure using pressure-probe techniques could provide a powerful tool to test thehypotheses put forward (Colombani & Forterre 2011).

    4.1.2. Bladderworts. Another nice example of plant movement involving the snapping of anelastic shell is provided by the bladderworts (Utricularia).The bladderworts represent a large andwidely distributed genus of rootless semi-aquatic plants that circumvent nutrient shortages bycatching tiny creatures (microorganisms, small arthropods) in water or on swampy ground (Lloyd1942, Juniper et al. 1989) (Figure 10d ). The traps consist of small (0.55-mm) bladders filledwith water and closed by a trapdoor. Capture is based on a suction mechanism (Figure 10d ).

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    During a slow initial phase, water is pumped outside the bladder so that elastic energy is storedin the trap body as it flattens and curves inward, which results in a decrease of water pressure inthe trap (pressure difference !P = 1015 kPa) (Sydenham & Findlay 1973, Sasago & Sibaoka1985, Singh et al. 2011). When the prey stimulates the trigger hairs located at the base of thetrapdoor, the trapdoor suddenly opens inward, and the stored elastic energy in the bladder israpidly converted into kinetic energy as water is sucked inside the bladder together with the prey.The whole trapping sequence lasts approximately 3 ms, with a suction phase of only 0.5 ms (Singhet al. 2011, Vincent et al. 2011). This timescale may be understood using a simple spring/massargument, in which the spring stiffness is related to the trap body elasticity and the mass is givenby the volume of displaced water (Vincent et al. 2011).

    The opening mechanism of the trapdoor itself has been partly unveiled in a recent study(Vincent et al. 2011) (Figure 10e, f ). In the set condition, the trapdoor has a is a shallow domewhose convex face is facing outward, thereby resisting the pressure difference across it much likea stone arch. Triggering induces a buckling transition in the trapdoor, which rapidly reverses itscurvature from convex to concave. In this new configuration, the door is no longer able to sustainthe pressure difference and rapidly swings inward. As for the Venus flytrap, the mechanism bywhich the buckling onset is reached is not fully elucidated. However, it is interesting to notethat, without any prey, the trap fires and resets periodically, which could be the signature of aspontaneous buckling once the pressure drop reaches a critical threshold (Vincent et al. 2011).

    4.2. Cavitation Catapults and Squirt GunsSome of the fastest movements on record involve a relatively slow storage of elastic energy withincell walls and the rapid release of this energy by some sort of fracture. The squirt gun of thefungus Pilobolus is one elegant example (Figure 11) (Yafetto et al. 2008). Morphologically, Pi-lobolus resembles Phycomyces discussed in Section 3.1. As in Phycomyces, the long sporangiophoresupports a terminal sporangium that must be dispersed (Figure 11a). The distal end of the stalkis swollen, and its wall is endowed with great elasticity. While the terminal sporangium matures,the stalk cell below starts swelling and building elastic energy. Although the pressure of 0.55 MPa

    Spore

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    a b ct = 0

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    100 m

    t = 40 s t = 80 s t = 120 s t = 160 s t = 200 s

    Stalkcell

    Sporangium

    Figure 11Squirt guns and water cohesion catapults for the dispersal of spores. (a) The Pilobolus sporangiophore comprises a terminal sporangiumfilled with spores (black) and is supported by a large stalk cell that stores elastic energy in its wall (blue). (b) The first 200 s of theejection. Note the collapse and recoil of the stalk cell as it squirts out its cytoplasmic content. (c) Spore ejection in the fernleptosporangium. A row of specialized cells known as the annulus is the motor of the catapult. Water loss by the annulus forces thelateral walls of the annular cells to approach each other, leading to an inversion of the sporangium. The water tension that builds in theannular cells is formidable (as much as 30 MPa) and will ultimately lead to cavitation of one cell, setting a chain reaction across theannulus. At this point, the sporangium shuts back into its closed position, releasing the spores in the process.

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    (Yafetto et al. 2008) is not exceptionally high, the stresses it generates in the wall are (>5.5 MPaor 55 bars). The concomitant increase in wall stresses and the weakening of the contact regionbetween the stalk and sporangium lead to the fracture of the wall and the explosive release of thesporangium in approximately 0.05 ms. Here the characteristic time for the ejection is likely set bythe inertial time required to accelerate the water contained in the stalk (Figure 11b).

    One of the most elegant trigger mechanisms for this class of fast motions relies on the cohesiveproperty of water (Koller & Scheckler 1986). The tensile strength of a water column at roomtemperature exceeds 26 MPa (Briggs 1950), providing ample room for energy storage and releaseonce cavitation occurs. A number of organisms use the coupling of water volume changes andelastic storage to achieve motion (King 1944, Hovenkamp et al. 2009). The best-studied exampleis the fern leptosporangium (Figure 11c). The slow opening of the sporangium that contains thespores comes from the bending of the annulus, a ring-shaped organmade of a single row of cuboidcells (King 1944). Upon exposure to a dry environment, these cells lose water and thus decreasetheir volume. Volume reduction is converted into bending energy because the cells walls havea unique U-shape, thin on one side and thick on the other (Figure 11c). This slow backwardbending builds elastic strain within the annulus walls that is balanced by a high negative pressurewithin the annulus cells. At a critical pressure of approximately35MPa (Renner 1915, Ursprung1915), cavitation occurs in the cells, causing a rapid closure of the sporangium that launches thespores into the air.

    Other examples abound, in particular the wide range of explosive seed pods that store elasticenergy as they dry until a fracture propagates along a predetermined path, thus catapulting theseeds (Swaine & Beer 1977). Many of these systems have minimal water content when triggeredand come closest to the ultimate physical limit for fast motion, that set by the speed of elasticwaves within solids (see Figure 5b) (Skotheim & Mahadevan 2005).

    5. RELATED FLUID MECHANICS PHENOMENA

    5.1. Surface TensionDriven Movements: Propulsion of Fungal Spores

    The Basidiomycetes, a large class of fungi that includes the generic white mushroom found insupermarkets, have evolved what may be the most economical mechanism to generate movement.These mushrooms actively disperse their spores using the surface tension of water as their onlysource of energy (Buller 1909, Ingold 1939, Turner&Webster 1991). The spores, known as ballis-tospores, are borne by the gills of mushroom caps and must be ejected from the gill surface beforebeing picked up by air currents (Figure 12a). The ejection process begins with the condensationof a water drop, known as Bullers drop, at the proximal end of the spore and the growth of a thinfilm of water on the spore (Figure 12b,c). When the drop reaches a critical size, it touches thewater film on the spore surface. At this point, the surface tension quickly pulls the drop onto thespore, thus creating the necessary momentum to detach the spore from the supporting sterigmaand to set it in motion.

    The energy available to eject the spore comes from the difference in surface energy before andafter the fusion of Bullers drop: !Ep = 4piR2D (1 RD/RD), where is the energy associatedwith the liquid-vapor interface, and RD and RD are the radii of the drop before and after fusion,respectively (Noblin et al. 2009). Assuming that all the freed energy serves to accelerate the dropand using the conservation of momentum between the drop and spore, we predict a spore velocityof 1.2 m s1, whereas the observed velocity is 0.8 m s1. The predicted velocity is surprisinglyaccurate given that energy loss has not been taken into account in the model. Finally, it is possibleto compute the time required for ejection by considering the magnitude of the capillary force

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    Spore

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    Figure 12(ac) Surface-tension propulsion of fungal spores. (df ) Viscoelastic trap in pitcher plants. (a) Section of a typical mushroom capshowing the gills and the location of the spore-bearing basidia (insert). The approximate trajectory of the spore is shown as a dottedline. (b) Early stages of spore discharge in Auricularia auricula [250,000 frames per second (fps) and 1-s exposure]. (c) Four stages ofballistospore ejection, from drop growth and coalescence to ejection. The radius of the drop before (RD) and after (RD) coalescence isindicated. The circle-and-cross symbol indicates the position of the spores center of mass during the ejection process. (d ) Pitcher ofNepenthes rafflesiana (Brunei). (e) Dynamical sequence of a fly (Calliphora vomitoria) after falling into the digestive fluid, showing aviscoelastic liquid filament attached to its leg (arrows). Time between frames is 80 ms. ( f ) Capture rate of insects as a function of theDeborah number, defined as the ratio of the fluid elastic relaxation time to the typical half-period of the swimming stroke of insects inthe fluid . Panels ac taken from Noblin et al. (2009) and panels df from Gaume & Forterre (2007).

    RD, the inertia of the droplet, and the distance L the droplet must travel:

    c ap =(L3

    )1/2, (9)

    where L 2RD, and is the droplet density. We find that the ejection takes less than 1 s,making this movement the fastest to be completed.

    5.2. Viscoelastic Trap in Carnivorous Pitcher PlantsAs shown above, moving fast is an increasingly difficult challenge in plants as the system sizeincreases, and unsurprisingly the fastest motions have been found within the smallest systems.However, there are plenty of other strategies for the smart plant to cope with a mobile world. Onestrategy is the use of complex fluids, as recently discovered in some carnivorous plants.

    The tropical pitcher plants of the genusNepenthes (Figure 12d ) are among the most successfulcarnivorous plants in terms of prey spectra and the amount of insects trapped ( Juniper et al.1989). They have long been thought to function as simple pitfall traps relying on slippery surfaces

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    that decrease insect adhesion (Gaume et al. 2004, Gorb et al. 2005) and wettable surfaces thatcause insect aquaplaning (Bohn & Federle 2004). Recently, the digestive liquid contained insidethe pitcher has been shown to play a crucial role in the trapping mechanism as well (Gaume &Forterre 2007, Bonhomme et al. 2011). High-speed videos of flies thrown on the free surface ofthe digestive liquid reveal that they are unable to escape from the fluid, their legs being tetheredby sticky viscoelastic filaments typical of biological fluids composed of long-chain polymers suchas mucus or saliva (Figure 12e). Remarkably, the trapping efficiency of the digestive liquid remainhigh even when the fluid is highly diluted by water, a property of great adaptive significance forthese tropical plants, which are often subjected to heavy rainfalls.

    Measurements of the thinning dynamics of the liquid filaments [capillary break-up rheometry(Rodd et al. 2005)] show that this unique trapping property comes from the high viscoelasticityand apparent elongational viscosity of the digestive liquid, which may be more than 105 that ofwater. The capture/escape transition is actually controlled by the Deborah number De, defined asthe ratio between the elastic relaxation time of the liquid and the typical period of insect swimmingin the pool (Figure 12f ). Capture occurs for De > 1, when the elastic forces created by insectmovements have no time to relax (Gaume & Forterre 2007). This study shows how the use ofcomplex fluids provides an alternative to motion in some cases. The same strategy is likely appliedat a smaller scale in other carnivorous plants such asDrosera (sundews) or Pinguicula (butterworts),which have sticky leaves to immobilize their prey, before engulfing them slowly.

    6. CONCLUSIONMotility in plants is made possible by a few broad mechanisms that are variations on a commonthemethe interaction of water with a solid phase made of the cell walls. Although water-drivenmovements may lack the rapid temporal control of muscular movements, they offer some clearadvantages. A comparison with the peak force generated by animal tissue is illuminating. Tissuesspecialized for force generation such as muscle will at best produce a tensile stress of 0.3 MPa(Bray 2001). These stress levels are reached at the cost of filling muscle cells with cytoskeletalproteins and motors. In contrast, plants can achieve a much wider range of stresses (Figure 2a),both tensile and compressive, without the need for extensive cellular specialization beyond thepresence of the cell wall. Therefore, in terms of versatility and design potential, the plants strategyto generate movement is certainly appealing. Finally, the sole reliance on water and cellulose, themost abundant biopolymer on Earth, offers a clear and cheap path for biomimetic designs.

    In this review, we mostly focus on the physical mechanisms behind plant motion and onlyskim over the physiology and molecular signaling associated with these processes. Much remainsto be done to fill the gap between the physical and biological approaches. Future efforts shouldcombine tools and concepts from bothmechanics andmodern biology to provide a comprehensivedescription of these nonmuscular machines.

    SUMMARY POINTS

    1. From the slow growth of shoots and roots to the rapid snapping of carnivorous plants andthe explosive rupture of seed pods, the characteristic times of plant motion span almost10 orders of magnitude.

    2. Plants and other walled organisms such as fungi are hydraulic machines that rely onthe high differential hydrostatic pressure supported by their thin and stiff cell walls toproduce movements.

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    3. At the cellular level, hydraulic movements (reversible swelling/shrinking, irreversiblegrowth) are driven by osmotic or humidity gradients. Thematerial properties of cell walls(elasticity, plasticity, and anisotropy) and cell geometry set the timescale and amplitudeof motion.

    4. Growth (irreversible) movements are mainly controlled by the rheological properties ofthe cell wall, which is a complex yield-stress fluid under tight molecular and physiologicalcontrol.

    5. The poroelastic timescale of water diffusion at the cellular and tissue level constrainsthe maximal speed of purely hydraulic movements and provides a physical basis for theclassification of motion in plants and other walled organisms.

    6. To overcome the poroelastic limit and generate faster movements, plants couple passiveor activewater transport (osmosis, evaporation) tomechanical instabilities (snap buckling,cavitation, fracture).

    7. At the micrometer scale, the surface tension of water can serve as an energy store andtriggermechanism, without the need of an elastic solid phase. This mechanism is perhapsthe purest strategy to power locomotion and allows basidiomycete fungi to dispersemillions of spores cheaply.

    8. Complex viscoelastic fluids that can ensnare insects offer an alternative to motion incarnivorous plants.

    FUTURE ISSUES

    1. Plant movements result from the interplay between water transport and cell wall defor-mation and thus offer a wealth of opportunities for studies of fluid-structure interactions.An investigation of poroelasticity in complex materials mimicking plant tissues (porouscellular materials, slender geometries) would help us better understand the dynamics andconstraints on plantmovements. Coupling with other processes such as elasto-capillarity,cavitation, and evaporation should be investigated as well.

    2. An additional feature of living systems is growth. The constitutive modeling of complexfluids that add mass and remodel as they deform is an active area of research. Betterunderstanding the mechanical coupling between growth, osmotic pressure, and watertransport in cellular tissues is a great challenge.

    3. Although the kinematics and dynamics of plant motion are well understood at a macro-scopic level, the associated mechanical processes have been much less explored at thecellular and tissue level. There is a need for quantitative physics and mechanics mea-surements in cells and tissues (noninvasive flow visualization, cell force and pressuremeasurements) to better understand the link between the biological signals (osmoticgradients, action potential) and the mechanical responses (change in turgor pressure, cellwall softening) in plant motion.

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    4. Finally, movements in plants at themolecular level are poorly understood. Future studiesshould tackle the constraints brought by these movements on the timescale of importantphysiological processes such as flow in water channels and rapid ion transport across cellmembranes. In particular, it is still not clear whether the contractile properties of theactin and microtubule cytoskeleton have to be ruled out completely for plant movements(Morillon et al. 2001, Kanzawa et al. 2006).

    DISCLOSURE STATEMENTThe authors are not aware of any biases that might be perceived as affecting the objectivity of thisreview.

    ACKNOWLEDGMENTSWe thank J. Skotheim and P. Marmottant for supplying figures and data. Y.F. thanks the AgenceNationale pour la Recherche for financial support. J.D. thanks the Harvard Center for NanoscaleSystems for use of their microscope facility and the Materials Research Science and EngineeringCenter for financial support.

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    RELATED RESOURCESWeb sites on plant motion: http://plantsinmotion.bio.indiana.edu, http://www.snv.jussieu.

    fr/bmedia/mouvements/index.htm (in French)DVD on plant motion: Attenborough D. 1995. The Private Life of Plants. London: BBC. 390 min.Peters Savage Garden: http://www.exploratorium.edu/gardening/feed/peter_savage_

    garden/Charles Darwins The Power of Movement in Plants: http://darwin-online.org.uk/

    EditorialIntroductions/Freeman_ThePowerofMovementinPlants.htmlMovie of nematode-trapping fungi:http://archive.microbelibrary.org/ASMOnly/Details.asp?

    ID=1769Movie of bladderworts (Utricularia): http://www.youtube.com/watch?v=Zb_SLZFsMyQMovie of the fungus Pilobolus: http://www.plantpath.cornell.edu/PhotoLab/TimeLapse2/

    Pilobolus1_crop1_FC.html

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