The Role of Water in Plant Movements

FL44CH19-Forterre ARI 18 November 2011 15:49

“Vegetable Dynamicks”: The Role of Water in Plant Movements

Jacques Dumais1 and Yoel¨ Forterre2

1Department of Organismic and Evolutionary Biology, Harvard University, Cambridge, Massachusetts 02138; email: [email protected] 2IUSTI, CNRS, Aix-Marseille Universite,´ 13453 Marseille cedex 13, France; email: [email protected]

Annu. Rev. Fluid Mech. 2012. 44:453–78 Keywords The Annual Review of Fluid Mechanics is online at plant biomechanics, osmotic pressure, growth, poroelasticity, instability, fluid.annualreviews.org surface tension, complex fluids This article’s doi: by 82.245.254.175 on 12/23/11. For personal use only. 10.1146/annurev-fluid-120710-101200 Abstract Copyright c 2012 by Annual Reviews. ! Although they lack muscle, plants have evolved a remarkable range of mecha- All rights reserved nisms to create motion, from the slow growth of shoots to the rapid snapping 0066-4189/12/0115-0453$20.00 of carnivorous plants and the explosive rupture of seed pods. Here we review Annu. Rev. Fluid Mech. 2012.44:453-478. Downloaded from www.annualreviews.org 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 discuss movements that rely only on the transport of water, from irreversible growth to reversible swelling/shrinking due to osmotic or humidity gradients. 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.

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1. INTRODUCTION Plants 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 it to the leaves. The branching of the shoots and roots, the anastomosing venation of leaves, and the structural organization of plant tissues all speak to their transport function. Given this, it is not surprising that plants have attracted attention in applied fluid mechanics (Canny 1977, Rand 1983). The first detailed study of the movement of water within plants is provided by Hales’s (1727) “Vegetable Staticks” (Figure 1a). Hales (1727) not only described with great accuracy the transpiration stream of plants, he also sought to interpret his observations in light of the fluid mechanics of his time: “The sap vessels are so curiously adapted by their exceeding fineness, to raise 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 interest of 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 development and plants’ responses to environmental stimuli such as light and gravity (Gilroy & Masson 2008, Moulia & Fournier 2009). In engineering and applied sciences, these nonmuscular movements have provided inspiration for biomimetic design in the areas of microfluidics and robotics (Taya 2003, Burgert & Fratzl 2009, Martone et al. 2010). The goal of this review is to present some key fluid mechanics principles used by plants to achieve 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, for example, in the ascent of sap or the translocation of sugars in vascular systems. These topics constitute a broad field of research and have already been the subject of many monographs and reviews (Tyree & Zimmermann 2002, Holbrook & Zwieniecki 2005), including reviews in this series (Canny 1977, Rand 1983). Still the diversity of mechanisms that fit within the scope of this review is vast. From the slow growth 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 plants to trap prey or disperse their seeds (Sibaoka 1969, Hill & Findlay 1981). by 82.245.254.175 on 12/23/11. For personal use only.

ab Annu. Rev. Fluid Mech. 2012.44:453-478. Downloaded from www.annualreviews.org

Figure 1 Vegetable staticks and dynamicks. (a)ExperimentalsetupusedbyHales(1727)tocollectthewater transpired by various plants. (b) A recording of leaf movement by Darwin & Darwin (1880).

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We begin with a brief overview of the plant cell and tissue structure and a discussion of the various 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 various movements. In Section 3, we discuss hydraulic movements at the cell and tissue levels, from irreversible growth to reversible swelling/shrinking. In both cases, the driving force for motion is the high differential water pressure supported by the plant cell wall. In Section 4, we show how mechanical instabilities—snap buckling, cavitation, and fracture—coupled with water transport allow 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 involving capillarity and complex fluids.

2. FUNDAMENTALS A bouquet of tulips left standing without water will gradually droop, the flaccid stems unable to support the weight of the blossoms. If water is provided again, the stems will soon straighten, and the 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 their semipermeable membranes. The local changes in cellular volume and tissue stiffness enable the large-scale tissue deformations required for motion. Therefore, plant motion is fundamentally a mechanical problem whose key features are rooted in the structure and physiology of plant cells. Within the context of mechanics, the plant body can be decomposed into two simple phases: a fluid phase representing as much as 75% of the total mass of fresh tissue and a solid phase made largely of the cellulosic walls that surround every plant cell. At this level, plants do not differ significantly from algae, fungi, and other organisms with walled cells. We therefore have sampled freely between these groups to emphasize the generality of the principles at work, but also to highlight the great ingenuity with which nature has sought to endow these organisms with the power of movement.

2.1. Plant Cell and Tissue Structure by 82.245.254.175 on 12/23/11. For personal use only. A fundamental difference between plants and animals is that plant cells are surrounded by a thin but 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 muscle Annu. Rev. Fluid Mech. 2012.44:453-478. Downloaded from www.annualreviews.org fibers) to deform and generate movement. However, the stiff wall allows plant cells to sustain a large 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 pressure by maintaining an osmotic gradient between their cytoplasm and the environment, thus allowing water to move into the cell and put the wall under tension. As shown below, this high turgor pressure provides the force for changing the cell volume and is thus the main motor for growth and 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 as the volume contained within the plasma membrane of cells and thus under direct osmotic control by the cells. The apoplast is the dual of the symplast and includes all the volume taken by cell walls and intercellular spaces. Direct symplastic flow between cells is possible through intercellular

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ab Guard cell Plasma Phloem membrane

Typical plant cell Cell wall

Muscle 1

Car tire 3

Blood pressure 2 Xylem tension

Fern leptosporangium

–30 –25 –5 –4 –3 –2–10 1 2 3 4 Pressure (MPa)

Figure 2 Water 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 case − of the fern leptosporangium. (b)Theprincipalpathsalongwhichwaterﬂows.Thecell-to-cellpath! 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 (double blue line). In the apoplast path ", water moves through the relatively porous cell walls and intercellular spaces without crossing any membranes. Finally, evaporative loss at the plant surface # plays an important role in water relations and also drives many plant movements.

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

2.2. Water: The Prime Mover

by 82.245.254.175 on 12/23/11. For personal use only. The flow of water in living plant tissues shares many similarities with flow in porous media, with the 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 the pressure and osmotic gradients will vary slightly. Therefore, we consider in turn water flow at the cellular level and at the tissue level. Annu. Rev. Fluid Mech. 2012.44:453-478. Downloaded from www.annualreviews.org

2.2.1. Water transport across the cell membrane. At the cellular level, the water ﬂux j (in meters per second) across a perfect semipermeable membrane, and thus the rate of change in the cell volume V, is driven by the difference in the chemical potential of water, according to the relation dV jA AL !" AL ( !P !π), (1) dt = =− p = p − + Water potential: the where A is the cell membrane area; Lp is the hydraulic conductivity of the cell membrane (in meters chemical potential of per second per pascal); " P π is the water potential (in pascals); P is the hydrostatic pressure = − water per unit volume minus the atmospheric pressure; π c T is the osmotic component, with c the solute molar con- relative to a reference 1 1 = R centration, 8.32 J mol− K− the gas constant, and T the temperature (in Kelvin); and ! stands state R = for the difference between the inside and outside of the cell (Dainty 1976, Finkelstein 1987, Kramer

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& Boyer 1995, Nobel 1999). In the field of animal physiology, Equation 1 is known as the Starling equation and is used to quantify the rate of plasma filtration and resorption in capillary beds. In the context of plant cells, Equation 1 states that changes in the volume of cells are driven by mod- ifications of the turgor pressure or solute concentration from their equilibrium values. The rate of

water ﬂow depends on the membrane conductivity Lp, which is a crucial quantity for water relations 13 11 1 1 in plants. Values of the membrane conductivity span a wide range, 10− L 10− ms− Pa− ≤ p ≤ (Steudle 1989), and may be regulated by the opening or closing of 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 level apply to the description of flow in tissues at the macroscopic level. For instance, the assumption of only cell-to-cell transport across the cell membranes (path 1 in Figure 2b)yieldsaDarcy-likerela- tion between the water flux and the water-potential gradient, with a Darcy permeability (in squared meters) ηL R,whereη (in pascal seconds) is the water viscosity and R is the typical cell size (Philip ∼ p 1958). However, the situation in most tissues is more complex because water may also flow in the cell wall continuum (apoplast pathway), whose permeability φk depends on both the cell wall ∼ wall volume fraction φ in the tissue and the cell wall permeability kwall (Figure 2b). In many cases (roots and stems), both pathways are of similar conductivity, and one may define a single effective perme- 20 ability k encompassing both the cell-to-cell and apoplast pathways, with typical values 10− k 19 2 ≤ ≤ 10− m (Molz & Ikenberry 1974, Molz & Ferrier 1982, Steudle 1992). However, although osmotic and turgor gradients may drive water flow across the cell membrane, only pressure gradients lead to significant bulk flow in the apoplast because the cell wall continuum is permeable to most solutes (Steudle 1989, 1992). Thus both driving forces are not equivalent at the tissue level.

2.3. Material Properties of the Plant Cell Wall The plant cell wall is a living material and as such must be approached mechanically with some caution. For example, what may look like simple viscosity at the macroscopic level can in fact reﬂect the addition of mass and a chemically mediated remodeling of wall material at the microscopic level. The constitutive modeling of materials that add mass and remodel as they deform is an active area of research (Ambrosi et al. 2011). Here we focus mostly on the macroscopic properties of cell walls, as they sufﬁce in explaining plant movements. by 82.245.254.175 on 12/23/11. For personal use only. 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 (see

Section 3.2), in which higher wall strains can also be reversible. The Young’s modulus Ewall of the cell wall of actively expanding cells is typically less than 1 GPa (Probine & Preston 1962), whereas Annu. Rev. Fluid Mech. 2012.44:453-478. Downloaded from www.annualreviews.org for wood fibers, it exceeds 25 GPa (Gibson & Ashby 1999). An alternative measure of elasticity is the bulk modulus of the cell ε V (dP/dV )(Figure 3a). The bulk modulus characterizes how = changes in volume are related to changes in turgor pressure, and has a typical value between 1 and 50 MPa (Steudle et al. 1977, Cosgrove 1988). Although it is defined in the same way as the bulk modulus of standard engineering materials, the interpretation of the cellular bulk modulus can be challenging 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 as it 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 Young’s modulus of the tissue and the cell properties (wall elasticity, cell size) can be approached from the standpoint of cellular materials (Gibson & Ashby 1999, Gibson et al. 2010). As long as the turgor pressure is high and the cell walls stretched, the tissue Young’s modulus scales with the cell bulk modulus,

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abc 900 104 0.60 0.55 800

0.50 t (MPa)

P 700 0.45 /d

103 L 600 30

(µm) 500 102 L 20 400 300 10 101

Relative volume (%) volume Relative 200 Percent original d Percent 0 100

100 elongation Cell 0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0 20 40 60 80 100 120 0 50 100 150 Cell pressure P (MPa) Time (min) Percent original pressure

Figure 3 (a)Elasticand(b,c)plasticbehaviorofthecellwallasobservedinChara and Nitella internodal cells. (a) The pressure-volume curve is shown for a single internodal cell in the elastic regime (Kamiya et al. 1963). Note the hysteretic response. (b) Increases or decreases in turgor 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 cell’s pressure and shows a marked yield pressure (approximately 50% of the normal pressure) below which growth is not possible (Proseus et al. 2000).

E ε 10 MPa. However, E decreases sharply when the turgor pressure drops to zero because ∼ ∼ the 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 wall has been described as a Bingham fluid (Lockhart 1965, Green et al. 1971, Cosgrove 1985, Proseus et al. 1999). Bingham fluids differ from ordinary viscous fluids in that they deform irreversibly

only for stresses that exceed a plastic yield stress σ y. For uniaxial deformation, the rate of plastic deformation is written as (˙ χ(σ σ ) for σ>σ,whereχ is the material extensibility (an = − y y inverse viscosity). Lockhart (1965) was the ﬁrst to apply this constitutive law to wall extension and growth 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 are now available (Dumais et al. 2006, Dyson & Jensen 2010). by 82.245.254.175 on 12/23/11. For personal use only. The simple constitutive law adopted by Lockhart is well supported experimentally. A number of Tropic response: investigators have demonstrated that the relative rate of cell expansion is not simply proportional movement determined to the cell pressure (Green et al. 1971, Proseus et al. 1999). For example, small-step increases or by the direction of the decreases in pressure (<10%) cause drastic changes in the rate of cell expansion (Figure 3b). Also, Annu. Rev. Fluid Mech. 2012.44:453-478. Downloaded from www.annualreviews.org stimulus (gravity, light, with rare exceptions, cells show a clear yield pressure below which they cannot maintain growth touch); examples (Figure 3c). include phototropism, gravitropism, and thigmotropism 2.4. Timescales of Plant Movements Nastic response: movement that occurs As we have seen, plants mostly rely on water and their cellulosic wall to move. Given this, the in a direction diversity of movements and timescales observed in plants is quite remarkable. Comparing the slow independent of the growth of shoots (days to hours) with the explosive motion of seeds and spores (<0.1 ms), one stimulus (e.g., the sees that the characteristic times of plant motions span almost 10 orders of magnitude. Biologists folding of the Mimosa pudica leaf and closing usually classify these movements depending on their reversible or irreversible character, on their of the Venus ﬂytrap) active (physiological) or passive origin, or on the relation between the nature of the stimulus and the response (tropic or nastic response to light, touch, gravity, etc.) (Hill & Findlay 1981, Hart 1990,

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ab 1 25mM KCl 0.64 Pressure probe 1mM KCl 0

–1 (MPa) 0.63 Giant algal cell (Chara) P 2 0 5 10 15 20 25 Desiccation Time (min) 0.62 1 in dry air 0 0.61 –1

–2 0.60 Change in tissue length (%) length tissue in Change Cell turgor pressure pressure turgor Cell –3 Salt solution/ Stop air !ow Water bath fresh water –4 0.59 15 20 25 30 35 40 0 12345 Time (s) Time (min)

Figure 4 Swelling/shrinking timescales in plant cells and tissues. (a)Single-cellrelaxationinagiantalgalcell(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 small volume of water in the cell. In this system, V/A 200 µmandε 30 MPa. The cell relaxation time is τ 3.5s,whichgivesa = = cell = cell membrane conductivity L 210 12 ms 1 Pa 1.(b) Reversible swelling and shrinking of a plant tissue (pealed pea epicotyl 1 cm p ∼ − − − in 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 τ 40–80 s. In this radial water transport geometry, p = τ 0.022 L2/ (Philip 1958), where L 1.6 mm is the segment diameter, which gives a coefﬁcient of diffusion 10 9 m2 s 1. p = D = D ∼ − − Data taken from Cosgrove & Steudle (1981).

Braam 2005), with no clear reference to the underlying physical mechanism. A key suggestion made recently by Skotheim & Mahadevan (2005) is that the timescale for water transport constrains the maximal speed of hydraulic movements in these soft nonmuscular systems, thus providing a physical basis for the classiﬁcation of motion in plants and other walled organisms such as fungi.

by 82.245.254.175 on 12/23/11. For personal use only. 2.4.1. Cell relaxation time. We ﬁrst address this question at the cellular level. A cell perturbed from equilibrium by a small sudden change in the osmotic potential or turgor pressure relaxes exponentially with a timescale given by V R τcell , (2) Annu. Rev. Fluid Mech. 2012.44:453-478. Downloaded from www.annualreviews.org = A(ε !π)L ∼ εL + p p where R V/A is the mean cell radius, and ε !π, as is the case for most turgid cells (Dainty = $ 1976). Typical measurements in giant algal cells of R 200 µm, ε 30 MPa, and L p 12 1 1 = = = 2 10− ms− Pa− give τ 3.5s(Figure 4a). cell = The cell relaxation time (Equation 2) may be interpreted as the shortest response time of a plant cell to small water-potential perturbations and thus provides a bound for hydraulic movements at the cellular level. Figure 5a plots the timescale of motion τ as a function of the typical cell size R for a wide range of unicellular movements in plants and fungi, together with the boundary

given by Equation 2. Systems with τ>τcell can rely on water transport to swell or shrink, whereas systems with τ<τcell must use other mechanisms, as shown in Sections 4 and 5. The dependence

of τcell on the cell size R shows that water transport may be fast if a cell is sufﬁciently small, as in the case of the rapid swelling of nematode-trapping fungi (see Section 3.1).

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a Fastest cell swelling τcell b Fastest tissue swelling τp Fastest cell growth τLockhart Fastest elastic motion τel Nitella Lonicera japonica 6 Pollen tube 6 L. sempervirens 10 Sporangium (opening) 10 Pinguicula vulgaris Oedogonium Ipomoea nil Hydraulic movements Pharbitis nil 5 Stomata 5 10 Chara (relaxation) 10 (swelling/growth) Arabidopsis thaliana Arthrobotrys brochopaga (fungus) Oryza sativa Zoophagus insidians (fungus) Desmodium gyrans 104 A. dactyloides (fungus) 104 Mimosa pudica Dactylaria brochopaga (fungus) Dionaea Utricularia (trapdoor) Stylidium crossocephalum 103 Pilobolus 103 S. graminifolium Sporangium (closing) S. piliferum Ballistospore Utricularia (trapdoor) 102 102 Aldrovanda Catasetum

(s) (s) Lepidium campestre τ τ 1 1 Ruellia brittoniana 10 10 Ecballium elaterium Hura crepitans Arceuthobium 100 100

Timescale Timescale Mechanical 10–1 10–1 instability (buckling/cracks) 10–2 10–2

10–3 10–3

10–4 Ballistospores 10–4 (surface tension) 10–5 10–5 10–7 10–6 10–5 10–4 10–3 10–2 10–6 10–5 10–4 10–3 10–2 10–1 100 Cell radius R (m) Tissue size L (m)

Figure 5 Classification of (a)unicellularand(b) multicellular movements in plants and fungi, based on the timescale of water transport (from an original idea of Skotheim & Mahadevan 2005). The plots give the duration of movement τ as function of (a)thecellradiusR and (b)the tissue size L, defined as the smallest macroscopic moving part. The order of the labels in the figure key coincides with their order in the figure from top to bottom. In panel a, the solid blue line gives the cell relaxation time (the fastest cell swelling) τ R/(εL ), with cell = p ε 30 MPa and L 210 12 ms 1 Pa 1, and the dashed blue line gives the Lockhart time (the fastest cell growth not limited by = p = − − − water transport) τLockhart R/(!π L p ), with !π 0.5 MPa. In panel b,thesolidbluelinegivestheporoelastictime(thefastesttissue 2 = 9 2 = 1 swelling) τp 0.022 (L / ), with 10− m s− . The solid red line gives the timescale for elastic wave propagation (the fastest

by 82.245.254.175 on 12/23/11. For personal use only. = D D = elastic motion) τ L√ρ/E,withρ 1,000 kg m 3 and E 10 MPa. el = = − =

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 the

Annu. Rev. Fluid Mech. 2012.44:453-478. Downloaded from www.annualreviews.org transport of water from one part of the tissue to another. The ﬂow of water in a soft porous media such as plant tissue is a diffusive process, whose timescale over a tissue of size L,knownasthe

poroelastic time τ p, is given by

L2 ηL2 τ , (3) p ∼ ∼ kE D where kE/η is a diffusive coefﬁcient that depends on the Darcy permeability of the porous D = tissue k,the(drained)Young’smodulusofthetissueE, and the water viscosity η (Biot 1941, Wang 2000). Philip (1958) was the ﬁrst to derive a diffusion-like equation for water transport through plant tissues assuming cell-to-cell transport only. This model was further extended to incorporate both the cell-to-cell and the apoplast pathway (Molz & Ikenberry 1974, Molz & Ferrier 1982, Steu- 20 19 2 3 dle 1992). Typical values for plant tissues (k 10− /10− m , E 10 MPa, and η 10− Pa s) = = =

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10 9 2 give 10− /10− m , compatible with swelling experiments of tissues (Figure 4b)(Molz& D = Boyer 1978, Cosgrove & Steudle 1981, Steudle 1989).

The poroelastic time τ p (Equation 3) may be interpreted as the fastest possible water-driven movement at the tissue and organ levels (Skotheim & Mahadevan 2005). Its strong size dependence ( L2) shows that hydraulic movements are increasingly less efficient in terms of speed as the system ∝ size increases. Figure 5b presents the duration τ of plant movement as a function of the typical distance L through which the fluid is transported for a wide range of movements and timescales. The boundary τ τ naturally separates two categories of movements dominated either by = p swelling/growth or by mechanical instabilities. In the following, we present some specific examples of plant movement mechanisms in light of this classification. We first address slow hydraulic movements (irreversible growth, reversible swelling/shrinking). We then shift to rapid movements that cross the poroelastic boundary, in particular 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. It took the patient eye of Charles Darwin (Darwin & Darwin 1880) to ﬁnally draw greater attention to them. The elongation of stems is a slow but vital race toward the sun. Plants left behind are bound to spend their lives in the shadow of their taller neighbors. Given the selective pressure for rapid 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 water must enter the cell to increase its volume. Lockhart’s model for a cylindrical cell of radius R and wall thickness h predicts the following governing equation for the relative rate of volume increase: 1 dV (R!π hσ ) (˙ χ L − y . (4) = V dt = p hL χ R2 p + If typical values for cell geometry, hydraulic conductivity, and wall extensibility are considered, it 2

by 82.245.254.175 on 12/23/11. For personal use only. is seen that hL χ R . Lockhart’s equation thus simpliﬁes to p $ R!π (˙χ limited χ σy . (5) − = h − ! " This limit is known as the extensibility-limited regime for cell expansion. In other words, the Annu. Rev. Fluid Mech. 2012.44:453-478. Downloaded from www.annualreviews.org characteristic strain rate of growing plant cells is set by how fast the cell can extend its wall and not by its ability to take up water through the plasma membrane. Given that the extensibility χ is set by the rate at which the wall is synthesized and assembled by the cell, the extensibility-limited regime is really a statement about the cell’s maximal metabolic rate. In that context, it is a remarkable 1 observation that some cells can achieve local strain rates as high as 0.25 min− (Rojas et al. 2011) (an equivalent wall viscosity 1/χ 10 GPa s, assuming typical values R/h 100 and !π ∼ ∼ ∼ 0.5MPa). An elegant example of growth-mediated movement is seen in the phototropic response of the Phycomyces sporangiophore, a cell of gigantic size (as much as 10 cm) that raises the sporangium and its spores above the boundary layer for dispersal (Figure 6a). The zone of elongation is conﬁned to a 3-mm region just below the terminal sporangium. Within this region, strain rates 1 can reach values as high as 0.05 min− .Thehighstrainratesandrelativelynarrowsporangiophore

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ab 1 cm

1 mm 0 h 3 h 5 h 10 h 16 h 19 h

Figure 6 Growth movements at the cellular and tissue scales. (a) Phototropic bending of the Phycomyces sporangiophore 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. The coiling proceeds over several days until the terminal seed pod is buried into the ground.

1 stalk allow for rapid bending (3◦–7◦ min− ) in response to unidirectional light (Figure 6a) (Castle 1962). 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 to the maximum rate the membrane conductivity will allow (Equation 1). This regime is known as conductivity-limited growth (hL χ R2). In this limit, the Lockhart equation takes the form p ' (R!π hσy ) (˙L limited L p − . (6) p− = R2 Figure 5a shows the Lockhart timescale, defined as R 1 τLockhart . (7) = L p !π ∼ (˙L limited p− The Lockhart timescale may be interpreted as the fastest possible cell growth not limited by water diffusion across the membrane. It is usually much larger than the cell relaxation time (ε !π), $ except in cells in which the turgor pressure is very high and the wall is soft like stomata. by 82.245.254.175 on 12/23/11. For personal use only. One way cells reach the conductivity limit (Equations 6–7) is by assembling the wall surface necessary to increase cell volume before expansion. The algal cell Oedogonium offers a clear example (Figure 7a). Cell expansion in these algae is coupled to cell division and begins with the formation of an internal flap of wall material at one end of the cell. Because of its position within the external Annu. Rev. Fluid Mech. 2012.44:453-478. Downloaded from www.annualreviews.org load-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 transferring the load to the unstretched wall flap. Consequently, the initial phase of expansion is mainly limited by the rate at which water can enter the cell. Actively growing cultures of Oedogonium are described as twitching, providing some indication of the bursting nature of cell expansion following the release of the old wall. One observes in Figure 5a that Oedogonium’s growth rate is indeed close to the Lockhart limit. A similar mechanism has been put to good use in nematode-trapping fungi. These soil fungi have evolved a simple noose that they use to capture the nematodes with whom they share their environment (Figure 7c). Although the capture mechanism has not been fully elucidated, it is believed that a load-bearing wall layer loses its mechanical integrity upon stimulation of the noose cells by a passing worm. The ensuing sharp decline in turgor pressure puts the cell out of osmotic

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a b

Growth scar Wall ring Slit

10 µm c d

t = 0 s 5 µm t = 0.12 s t = 0.18 s 10 µm

Figure 7 Conductivity-limited growth. (a)ExpandinginnerwallringinOedogonium (Pickett-Heaps 1975). (b)Wall- ring development and early phase of expansion (Pickett-Heaps 1975). The wall ring develops inside the cell at the level of a slit in the load-bearing wall. Fracture of the outer wall along the slit puts the inner wall ring under tension, allowing for rapid cell elongation. (c) Constricting ring of the nematode-trapping fungus Arthrobotrys brochopaga.(d )Atrappednematode.Panelsc 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, leaving the 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 uptake rarely limits growth. However, in multicellular tissues, cells have access to water only through neighboring cells. One may ask whether, in that context, water ﬂow may not limit the rate of growth (Molz & Boyer 1978, Boyer & Silk 2004). Simple scaling arguments may be used to address this question. For uniform growth to occur in a plant tissue, this tissue must maintain a constant water ﬂux given by the Darcy law (k/η)!P/L,whereL is the typical size of the growing organ and

by 82.245.254.175 on 12/23/11. For personal use only. !P is the pressure difference. This ﬂux is related to the growth rate (˙ by volume conservation (k/η)!P/L L(˙. One may assume that the process is not conductivity limited as long as the ∼ pressure drop is small compared with the typical cell turgor P, yielding (˙ kP/ηL2.Theinverse ' of this maximal growth rate sets the tissue Lockhart timescale: 2 Annu. Rev. Fluid Mech. 2012.44:453-478. Downloaded from www.annualreviews.org ηL τ . (8) Lockhart tissue ∼ kP 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 these growth-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 might play a role in the regulation of growth movements (gravitropism, phototropism, thigmotropism).

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

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a 12

6 10 µm 1.0 MPa 2.0 MPa 3.0 MPa 4.0 MPa

2 Stomatal half-aperture (µm) Stomatal 50 µm Stomata 0 0123456 Guard cell turgor pressure P (MPa)

b 200 θ θ 150

Fast θ θ 100 Slow Angle Pulvinus Upper pulvinus 50 Proton signal (a.u.) Proton Lower pulvinus

0 0 200 400 600 800 1,000 Time (s)

Figure 8 Actuation in plants driven by osmotic gradients. (a)Relationshipbetweenthestomataapertureandthe turgor 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-leaf bean (Vicia faba). (b)Theslow recovery of the main pulvinus of Mimosa pudica imaged by nuclear magnetic resonance, showing a displacement of water from the upper to the lower half. The plot gives the distribution of water as a function of time, together with the petiole to stem angle. Figure adapted from Tamiya et al. (1988). by 82.245.254.175 on 12/23/11. For personal use only.

may be driven by an active transport of solutes (e.g., ions) across the cell membrane by specialized pumps. Although the variations in volume involved are usually small, the resulting change in turgor pressure may be very large as the cell wall is stiff (large bulk modulus). The additional coupling

Annu. Rev. Fluid Mech. 2012.44:453-478. Downloaded from www.annualreviews.org with the geometry and structural properties of the cell, tissue, or organ allows the amplifying mechanism to convert small reversible changes in volume into large movements. The opening and closing of stomata are probably among the most important and therefore best-studied reversible movements in plants (Meidner & Mansﬁeld 1968) (Figure 8a). Stomata are small pores on the surface of the leaves that control the leaf transpiration and gas exchange with 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 & Grams 2006). In most plants, the stomatal complex consists of two kidney-shaped guard cells that ﬂank a central pore (Taiz & Zeiger 2002). When the turgor pressure inside the guard cells increases owing to accumulation of solute (up to P 5 MPa), the cells swell reversibly (by 20%–40% in ∼ volume) and become more curved in shape, pushing apart the surrounding cells and opening the pore (Figure 8a). When the turgor pressure decreases, the two guard cells are pressed together

464 Dumais Forterre · FL44CH19-Forterre ARI 18 November 2011 15:49

by the surrounding cells, and the pore closes. The relationship between the stomatal aperture and the turgor pressure of the guard cells and the surrounding epidermal cells has been studied both experimentally 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 as they swell comes from the large asymmetry in wall thickness and the mechanical anisotropy of the cellulosic network. Other classic examples of actuation in plants are the circadian and light-induced leaf movements of several plants due to a specialized multicellular motor organ, called the pulvinus, located at the base of the leaf petiole (i.e., where the leaf comes in contact with the stem). These movements are usually under slow diurnal rhythms but can be as fast as seconds in the case of the leaves of the sensitive plant Mimosa pudica. The current paradigm attributes all pulvinus movements to a differential osmotic swelling/shrinking on opposite sides of the pulvinus, whose location at the base of the leaf acts as a lever to amplify the leaf’s angular motion (Hill & Findlay 1981, Moran 1990) (Figure 8b). The rapid closing of M. pudica seems to have attained the fastest possible hydraulic motion (Figure 5b) according to the size of its pulvinus [pulvinus diameter L 200 µm ∼ (Weintraub 1952), poroelastic time τ 1 s]. However, to our knowledge, no direct evidence for b ∼ rapid water ﬂow in this system exists.

3.3. Passive Actuation Driven by Humidity Gradients Osmotic 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, evaporation causes the cell volume to change as well. Many passive motions in plants are driven by these humidity (water-potential) gradients between the cell and the ambient air. The water po-

tential of air is a function of the water-vapor partial pressure Pvap through the classical van’t Hoff 3 1 relation " ( T /V¯ w) ln[P /P (T )], where V¯ w 18 cm mol− is the partial molar vol- vap = R vap sat ≈ ume of liquid water and Psat(T )thesaturationwaterpressure(Atkins&dePaula2002).Fora relative humidity 100 P /P (T ) 50%, the water potential in air is very low (approximately × vap sat = 94 MPa) and, together with water cohesion properties, may lead to large negative pressure in − cells with stiff walls (see Section 4.2) and force other cells to crumple tightly to accommodate the lost volume. Pollen grains are routinely exposed to harsh osmotic environments and have therefore evolved by 82.245.254.175 on 12/23/11. For personal use only. a coping mechanism. The surface of pollen grains comprises two regions: the porous apertures and the water-impermeable interapertural areas (Figure 9a,b). The permeability of the apertures is necessary to allow communication and exchange of water with the receptive surfaces of ﬂowers. The same permeability, however, threatens the survival of the minute pollen grains while in transit Annu. Rev. Fluid Mech. 2012.44:453-478. Downloaded from www.annualreviews.org from one ﬂower to another. The solution found by pollen grains is to design their apertures such that 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 fully reversible when water becomes available again.

3.4. Hygroscopic Movements Humidity-driven movements also occur in dead cells at the tissue level (sclerenchymal tissue) ( Jost & Gibson 1907). Sclerenchymal tissue typically consists of fiber cells with walls comprising several layers of oriented cellulose fibrils. When absorbing/expelling water in response to changes in air humidity, the tissue expands/shrinks anisotropically, perpendicular to the fibrils’ orientation (Fahn &Werker1972,Burgert&Fratzl2009).Asymmetryintheorientationofthefibrilsattheorgan

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a b

c d

e 5 f 50 6 40 7 θ θ 8 30

9 20 10 Number of turns Opening angle 11 10

12 0 0 400 800 1,200 0 100 200 300 400 500 t (s) t (min)

Figure 9 Passive movements driven by changes in humidity in (a,b)livingcellsand(c–f )deadtissues.Thefolding (self-sealing) response of (a)lilyand(b) Euphorbia pollen grains. (c,e)Thedrillingmotionoftheawnof Erodium (Evangelista et al. 2011). (e) Number of turns of the awn as a function of time for wetting and drying transients. The wetting time constant is τ 330 s, and the drying time constant is τ 400 s. (d,f ). The = = opening motion of conifer cones, showing much longer time constants in accordance with the larger system (Reyssat & Mahadevan 2009).

by 82.245.254.175 on 12/23/11. For personal use only. 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 & Mahadevan 2009) (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 these

Annu. Rev. Fluid Mech. 2012.44:453-478. Downloaded from www.annualreviews.org actuation mechanisms and those seen previously with living cells is that, in dead tissues, there are no longer cell membranes to maintain an osmotic gradient. Still, in some cases, a turgor gradient may be maintained at equilibrium with the atmosphere with the cell wall acting as a semipermeable membrane between the vapor and liquid phase (Wheeler & Stroock 2008). We note that a related mechanism of anisotropic swelling/shrinking driven by the cell wall architecture applies in the control of branch movements in woody organs (reaction wood), although on a different timescale and length scale (Niklas 1992).

4. BEYOND WATER DIFFUSION: INSTABILITY AND FLUID-SOLID COUPLING The hydraulic movements discussed above are constrained in terms of speed by the timescale of water diffusion τ ηL2/kE (poroelastic time), especially so as the system size increases (see p =

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Section 2). Yet, from carnivorous plants to seed dispersal, many plant motions cross this hydraulic boundary 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 water ﬂow—driven by either osmotic or humidity gradients—ﬁrst slowly stores elastic energy in the cell walls, 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 these champions of the plant kingdom, in light of recent discoveries made possible by the use of high- speed video. We note that the speed of these elastic movements is expected to be limited, ultimately, by the speed of elastic waves (Skotheim & Mahadevan 2005). The timescale τ L√ρ/E for el = elastic waves to travel a distance L,whereρ is the tissue density, is given in Figure 5b. No plant movements are found below this limit.

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

4.1.1. Venus flytrap. The rapid closure of the Venus flytrap (Dionaea muscipula) is one of the best-known rapid motions in the plant kingdom and led Darwin (1875) to describe the plant as “one of the most wonderful in the world” (Lloyd 1942, Juniper et al. 1989) (Figure 10a). Closure of the trap is initiated by the mechanical stimulation of one of the trigger hairs [usually twice within 20 s (Brown 1916)], which elicits an electrical action potential that spreads over the leaf in less than 1 s (Burdon-Sanderson 1882, Stuhlman & Darder 1950, Sibaoka 1969, Hodick & Sievers 1989, Volkov et al. 2007). The trap then shuts in a few tenths of a second, which seems too fast to be accounted for by pure water transport across the 0.5-mm leaf thickness (Figure 5 ). by 82.245.254.175 on 12/23/11. For personal use only. b ∼ 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 lobes of 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 perpendicular Annu. Rev. Fluid Mech. 2012.44:453-478. Downloaded from www.annualreviews.org to the midrib. However, because of the geometric constraint of the doubly curved lobe, this active bending causes the trap to accumulate stretching energy, until the stored elastic energy becomes so 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 inertial snapping of an elastic shell. It has been proposed that, within the hydrated soft porous tissue of the leaf, passive flow induced by bending provides viscous resistance that balances the elastic energy, in agreement with a poroelastic shell dynamics model (Figure 10c)(Forterreetal.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 at a macroscopic level, the mechanisms by which the plant actively changes its natural curvature and bend are still a matter of debate (von Guttenberg 1925, Lloyd 1942, Hill & Findlay 1981,

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a c 0.04

) 1.5 –3 0.02 1.0 × 10

–2

) 0.5 –1

(mm 0 g

0 κ

(mm –0.5

1 cm m 0.3 0.5 0.7 0.9 1.1 κ t (s) b –0.02

–0.04

Δt = 0.04 s –0.1 0.1 Local mean 0.3 0.5 0.7 0.9 1.1 –1 curvature κm (mm ) t (s)

d e Bladder Trapdoor ∆p Trapdoor

0.5 mm Threshold Slow 0.5 mm de!ation f

Fast suction

Figure 10 Snap buckling in carnivorous plants: (a–c)Venusflytrapand(d–f ) bladderworts. (a)Venusflytrapinthe(right panel ) open and (left panel )closedstate.(b) Dynamic sequence of the leaf closure measured using three-dimensional reconstruction. (c)Thespatially by 82.245.254.175 on 12/23/11. For personal use only. averaged mean curvature κm (blue) and the spatially averaged Gaussian curvature κg (red ) as a function of time (the trap was triggered at t 0). The solid line corresponds to the poroelastic theoretical model. (d ) Sketch of a bladderwort’s bladder (side view) and top view = of Utricularia inflata in the deflated ready-to-catch state and just after triggering. (e)Inversionofthetrapdoorandbucklingofthe median door axis visualized by light sheet fluorescence microscopy. ( f ) Numerical simulation of the trapdoor opening. Panels a–c taken from Forterre et al. (2005) and panels d–f from Vincent et al. (2011). Annu. Rev. Fluid Mech. 2012.44:453-478. Downloaded from www.annualreviews.org

Williams & Bennett 1982, Hodick & Sievers 1989, Volkov et al. 2008). Direct measurements of the cell turgor pressure using pressure-probe techniques could provide a powerful tool to test the hypotheses put forward (Colombani & Forterre 2011). Cell pressure probe: measures the turgor 4.1.2. Bladderworts. Another nice example of plant movement involving the snapping of an pressure in single plant elastic shell is provided by the bladderworts (Utricularia).The bladderworts represent a large and cells by means of a widely distributed genus of rootless semi-aquatic plants that circumvent nutrient shortages by microcapillary catching tiny creatures (microorganisms, small arthropods) in water or on swampy ground (Lloyd connected to a pressure gauge 1942, Juniper et al. 1989) (Figure 10d ). The traps consist of small (0.5–5-mm) bladders ﬁlled with water and closed by a trapdoor. Capture is based on a suction mechanism (Figure 10d ).

468 Dumais Forterre · FL44CH19-Forterre ARI 18 November 2011 15:49

During a slow initial phase, water is pumped outside the bladder so that elastic energy is stored in the trap body as it flattens and curves inward, which results in a decrease of water pressure in the trap (pressure difference !P 10–15 kPa) (Sydenham & Findlay 1973, Sasago & Sibaoka = 1985, Singh et al. 2011). When the prey stimulates the trigger hairs located at the base of the trapdoor, the trapdoor suddenly opens inward, and the stored elastic energy in the bladder is rapidly 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 (Singh et al. 2011, Vincent et al. 2011). This timescale may be understood using a simple spring/mass argument, in which the spring stiffness is related to the trap body elasticity and the mass is given by 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 is a shallow dome whose convex face is facing outward, thereby resisting the pressure difference across it much like a stone arch. Triggering induces a buckling transition in the trapdoor, which rapidly reverses its curvature from convex to concave. In this new configuration, the door is no longer able to sustain the pressure difference and rapidly swings inward. As for the Venus flytrap, the mechanism by which the buckling onset is reached is not fully elucidated. However, it is interesting to note that, without any prey, the trap fires and resets periodically, which could be the signature of a spontaneous buckling once the pressure drop reaches a critical threshold (Vincent et al. 2011).

4.2. Cavitation Catapults and Squirt Guns Some of the fastest movements on record involve a relatively slow storage of elastic energy within cell walls and the rapid release of this energy by some sort of fracture. The squirt gun of the fungus Pilobolus is one elegant example (Figure 11) (Yafetto et al. 2008). Morphologically, Pi- lobolus resembles Phycomyces discussed in Section 3.1. As in Phycomyces,thelongsporangiophore supports a terminal sporangium that must be dispersed (Figure 11a). The distal end of the stalk is 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

abt = 0 t = 40 µs t = 80 µs t = 120 µs t = 160 µs t = 200 µs c Sporangium by 82.245.254.175 on 12/23/11. For personal use only. Annulus

1 mm Spore Annu. Rev. Fluid Mech. 2012.44:453-478. Downloaded from www.annualreviews.org Stalk cell

100 µm

Figure 11 Squirt guns and water cohesion catapults for the dispersal of spores. (a)ThePilobolus sporangiophore comprises a terminal sporangium ﬁlled with spores (black) and is supported by a large stalk cell that stores elastic energy in its wall (blue). (b)Theﬁrst200µsofthe ejection. Note the collapse and recoil of the stalk cell as it squirts out its cytoplasmic content. (c) Spore ejection in the fern leptosporangium. A row of specialized cells known as the annulus is the motor of the catapult. Water loss by the annulus forces the lateral walls of the annular cells to approach each other, leading to an inversion of the sporangium. The water tension that builds in the annular cells is formidable (as much as 30 MPa) and will ultimately lead to cavitation of one cell, setting off a chain reaction across the − annulus. 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 MPa or 55 bars). The concomitant increase in wall stresses and the weakening of the contact region between the stalk and sporangium lead to the fracture of the wall and the explosive release of the sporangium in approximately 0.05 ms. Here the characteristic time for the ejection is likely set by the 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 cohesive property of water (Koller & Scheckler 1986). The tensile strength of a water column at room temperature exceeds 26 MPa (Briggs 1950), providing ample room for energy storage and release once cavitation occurs. A number of organisms use the coupling of water volume changes and elastic storage to achieve motion (King 1944, Hovenkamp et al. 2009). The best-studied example is the fern leptosporangium (Figure 11c). The slow opening of the sporangium that contains the spores comes from the bending of the annulus, a ring-shaped organ made of a single row of cuboid cells (King 1944). Upon exposure to a dry environment, these cells lose water and thus decrease their volume. Volume reduction is converted into bending energy because the cells’ walls have a unique U-shape, thin on one side and thick on the other (Figure 11c). This slow backward bending builds elastic strain within the annulus walls that is balanced by a high negative pressure within the annulus cells. At a critical pressure of approximately 35 MPa (Renner 1915, Ursprung − 1915), cavitation occurs in the cells, causing a rapid closure of the sporangium that launches the spores into the air. Other examples abound, in particular the wide range of explosive seed pods that store elastic energy as they dry until a fracture propagates along a predetermined path, thus catapulting the seeds (Swaine & Beer 1977). Many of these systems have minimal water content when triggered and come closest to the ultimate physical limit for fast motion, that set by the speed of elastic waves within solids (see Figure 5b)(Skotheim&Mahadevan2005).

5. RELATED FLUID MECHANICS PHENOMENA

5.1. Surface Tension–Driven Movements: Propulsion of Fungal Spores The Basidiomycetes, a large class of fungi that includes the generic white mushroom found in supermarkets, 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 only by 82.245.254.175 on 12/23/11. For personal use only. source of energy (Buller 1909, Ingold 1939, Turner & Webster 1991). The spores, known as ballistospores, are borne by the gills of mushroom caps and must be ejected from the gill surface before being picked up by air currents (Figure 12a). The ejection process begins with the condensation of a water drop, known as Buller’s drop, at the proximal end of the spore and the growth of a thin Annu. Rev. Fluid Mech. 2012.44:453-478. Downloaded from www.annualreviews.org ﬁlm of water on the spore (Figure 12b,c). When the drop reaches a critical size, it touches the water ﬁlm on the spore surface. At this point, the surface tension quickly pulls the drop onto the spore, thus creating the necessary momentum to detach the spore from the supporting sterigma and to set it in motion. The energy available to eject the spore comes from the difference in surface energy before and 2 after the fusion of Buller’s drop: !E 4π R γ (1 R /R* ), where γ is the energy associated p = D − D D with the liquid-vapor interface, and RD and R*D 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 drop and using the conservation of momentum between the drop and spore, we predict a spore velocity 1 1 of 1.2 m s− , whereas the observed velocity is 0.8 m s− . The predicted velocity is surprisingly accurate given that energy loss has not been taken into account in the model. Finally, it is possible to compute the time required for ejection by considering the magnitude of the capillary force

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a d e

1 cm Spore 1 cm Gill Basidium b 10 µm

1 cm

f 100

0 µs 4 µs 8 µs 12 µs 80

c Drop Early Late Ejection 60 growth coalescence coalescence Spore 40 R'D Film +

+ (%) rate Capture + + 20 Flies + + R Ants +D 0 0.1 1.0 10.0 Drop Deborah number, De = λ/τ

Figure 12 (a–c)Surface-tensionpropulsionoffungalspores.(d–f ) Viscoelastic trap in pitcher plants. (a)Sectionofatypicalmushroomcap showing the gills and the location of the spore-bearing basidia (insert). The approximate trajectory of the spore is shown as a red dotted line. (b) Early stages of spore discharge in Auricularia auricula [250,000 frames per second (fps) and 1-µs exposure]. (c)Fourstagesof ballistospore ejection, from drop growth and coalescence to ejection. The radius of the drop before (RD)andafter(R*D)coalescenceis indicated. The circle-and-cross symbol indicates the position of the spore’s center of mass during the ejection process. (d ) Pitcher of Nepenthes rafflesiana (Brunei). (e) Dynamical sequence of a fly (Calliphora vomitoria) after falling into the digestive fluid, showing a viscoelastic liquid filament attached to its leg (arrows). Time between frames is 80 ms. ( f )Capturerateofinsectsasafunctionofthe Deborah number, defined as the ratio of the fluid elastic relaxation time λ to the typical half-period of the swimming stroke of insects in the fluid τ.Panelsa–c taken from Noblin et al. (2009) and panels d–f from Gaume & Forterre (2007). by 82.245.254.175 on 12/23/11. For personal use only.

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

1/2 ρL3 τ , (9) cap = γ

Annu. Rev. Fluid Mech. 2012.44:453-478. Downloaded from www.annualreviews.org ! " where L 2R ,andρ is the droplet density. We ﬁnd that the ejection takes less than 1 µs, ≈ D making this movement the fastest to be completed.

5.2. Viscoelastic Trap in Carnivorous Pitcher Plants As shown above, moving fast is an increasingly difﬁcult challenge in plants as the system size increases, 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. One strategy is the use of complex ﬂuids, as recently discovered in some carnivorous plants. The tropical pitcher plants of the genus Nepenthes (Figure 12d )areamongthemostsuccessful carnivorous 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 that cause insect aquaplaning (Bohn & Federle 2004). Recently, the digestive liquid contained inside the 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 of the digestive liquid reveal that they are unable to escape from the fluid, their legs being tethered by sticky filaments typical of biological fluids composed of long-chain polymers such as mucus or saliva (Figure 12e). Remarkably, the trapping efficiency of the digestive liquid remains high even when the fluid is highly diluted by water, a property of great adaptive significance for these 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 viscoelasticity and apparent elongational viscosity of the digestive liquid, which may be more than 105 that of water. The capture/escape transition is actually controlled by the Deborah number De, defined as the ratio between the elastic relaxation time of the liquid and the typical period of insect swimming in the pool (Figure 12f ). Capture occurs for De > 1, when the elastic forces created by insect movements have no time to relax (Gaume & Forterre 2007). This study shows how the use of complex fluids provides an alternative to motion in some cases. The same strategy is likely applied at a smaller scale in other carnivorous plants such as Drosera (sundews) or Pinguicula (butterworts), which have sticky leaves to immobilize their prey, before engulfing them slowly.

6. CONCLUSION Motility in plants is made possible by a few broad mechanisms that are variations on a common theme—the interaction of water with a solid phase made of the cell walls. Although water-driven movements may lack the rapid temporal control of muscular movements, they offer some clear advantages. A comparison with the peak force generated by animal tissue is illuminating. Tissues specialized 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 cytoskeletal proteins 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 the presence of the cell wall. Therefore, in terms of versatility and design potential, the plants’ strategy to generate movement is certainly appealing. Finally, the sole reliance on water and cellulose, the

by 82.245.254.175 on 12/23/11. For personal use only. most 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 only skim over the physiology and molecular signaling associated with these processes. Much remains to be done to ﬁll the gap between the physical and biological approaches. Future efforts should

Annu. Rev. Fluid Mech. 2012.44:453-478. Downloaded from www.annualreviews.org combine tools and concepts from both mechanics and modern biology to provide a comprehensive description of these nonmuscular machines.

SUMMARY POINTS 1. From the slow growth of shoots and roots to the rapid snapping of carnivorous plants and the explosive rupture of seed pods, the characteristic times of plant motion span almost 10 orders of magnitude. 2. Plants and other walled organisms such as fungi are hydraulic machines that rely on the high differential hydrostatic pressure supported by their thin and stiff cell walls to produce movements.

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3. At the cellular level, hydraulic movements (reversible swelling/shrinking, irreversible growth) are driven by osmotic or humidity gradients. The material properties of cell walls (elasticity, plasticity, and anisotropy) and cell geometry set the timescale and amplitude of motion. 4. Growth (irreversible) movements are mainly controlled by the rheological properties of the cell wall, which is a complex yield-stress fluid under tight molecular and physiological control. 5. The poroelastic timescale of water diffusion at the cellular and tissue levels constrains the maximal speed of purely hydraulic movements and provides a physical basis for the classification of motion in plants and other walled organisms. 6. To overcome the poroelastic limit and generate faster movements, plants couple passive or active water transport (osmosis, evaporation) to mechanical instabilities (snap buckling, cavitation, fracture). 7. At the micrometer scale, the surface tension of water can serve as an energy store and trigger mechanism, without the need of an elastic solid phase. This mechanism is perhaps the purest strategy to power locomotion and allows basidiomycete fungi to disperse millions of spores cheaply. 8. Complex viscoelastic fluids that can ensnare insects offer an alternative to motion in carnivorous plants.

FUTURE ISSUES 1. Plant movements result from the interplay between water transport and cell wall deformation and thus offer a wealth of opportunities for studies of ﬂuid-structure interactions. An investigation of poroelasticity in complex materials mimicking plant tissues (porous

by 82.245.254.175 on 12/23/11. For personal use only. cellular materials, slender geometries) would help us better understand the dynamics and constraints on plant movements. 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 complex

Annu. Rev. Fluid Mech. 2012.44:453-478. Downloaded from www.annualreviews.org ﬂuids that add mass and remodel as they deform is an active area of research. Better understanding the mechanical coupling between growth, osmotic pressure, and water transport in cellular tissues is a great challenge. 3. Although the kinematics and dynamics of plant motion are well understood at a macroscopic level, the associated mechanical processes have been much less explored at the cellular and tissue levels. There is a need for quantitative physics and mechanics measurements in cells and tissues (noninvasive ﬂow visualization, cell force and pressure measurements) to better understand the link between the biological signals (osmotic gradients, action potential) and the mechanical responses (change in turgor pressure, cell wall softening) in plant motion.

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

DISCLOSURE STATEMENT The authors are not aware of any biases that might be perceived as affecting the objectivity of this review.

ACKNOWLEDGMENTS We thank J. Skotheim and P. Marmottant for supplying figures and data. Y.F. thanks the Agence Nationale pour la Recherche for financial support. J.D. thanks the Harvard Center for Nanoscale Systems for use of its microscope facility and the Materials Research Science and Engineering Center for financial support.

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RELATED RESOURCES Web 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.390min. Peter’s Savage Garden: http://www.exploratorium.edu/gardening/feed/peter_savage_ garden/ Charles Darwin’s “The Power of Movement in Plants”: http://darwin-online.org.uk/ EditorialIntroductions/Freeman_ThePowerofMovementinPlants.html

by 82.245.254.175 on 12/23/11. For personal use only. Movie of nematode-trapping fungi: http://archive.microbelibrary.org/ASMOnly/Details.asp? ID 1769 = Movie of bladderworts (Utricularia): http://www.youtube.com/watch?v Zb_SLZFsMyQ = Movie of the fungus Pilobolus: http://www.plantpath.cornell.edu/PhotoLab/TimeLapse2/ Pilobolus1_crop1_FC.html Annu. Rev. Fluid Mech. 2012.44:453-478. Downloaded from www.annualreviews.org

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