Chapter 1: Basic Soft Matter for Plants

Plants manipulate and move water to perform almost all their physiological functions. Yet, there is no microscopic “water pump” in plants that would “actively” transport water molecules across cells, as there is for proteins or ions. Therefore, whether minute transmembrane flows or large bulk transport through trees, all water movement in plants occurs passively, according to the well-known principles of thermodynamics and fluid mechanics. In this section, we review those mechanisms and present the main physical laws of water transport in plants. Readers interested in a more comprehensive presentation of both the biological and physical contexts may wish to consult recent reviews on the topic, such as Stroock et al.¹⁵³  or Jensen et al.⁷⁵ 

In plant science, it is common to characterize the thermodynamic status of water using water potential Ψ, defined as the chemical potential of water relative to a reference state, per unit volume:^{35,85,110,112} 

Here μ_w is the chemical potential of water, i.e. the Gibbs free energy per unit mole, μ$w0$ is the chemical potential of water in the reference state, chosen to be pure liquid at atmospheric pressure and at ambient temperature, and ν_w ≈ 18 cm³ mol⁻¹ is the molar volume of liquid water. Using this definition, the water potential has the dimension of a pressure (unit Pa or J m⁻³) and is null for pure water at 1 atmosphere. The water potential depends on the thermodynamic state of water, i.e. its solid, liquid or gaseous state, the presence of other molecular constituents interacting with the water and the possible presence of an external field such as gravity acting on the water. We provide below the main expressions of water potential useful for plants.

• For a diluted aqueous solution under gravity, as found in the vessels of the vascular system or in the large vacuoles of cells (Box 1.1), the water potential may be decomposed as:

The first term Ψ_p = P is the enthalpic or pressure contribution to the water potential³, where P is defined as the total pressure of the solution minus the atmospheric pressure. When referring to the pressure inside plant cells, this pressure difference is usually called the turgor pressure. The second term Ψ_os = −cT is the osmotic contribution corresponding to the free enthalpy of mixing of the solute in water, as given by the van't Hoff's law. Here c is the solute concentration (unit mol m⁻³),  ≃ 8.31 JK⁻¹ mol⁻¹ is the ideal gas constant and T is the absolute temperature (unit Kelvin), such that T ≃ 2500 J mol⁻¹ at ambient temperature. The opposite of the osmotic term Π = −Ψ_os = cT, thereafter called the osmotic pressure, is often used instead of Ψ_os. Finally the last term Ψ_g = ρgz is the contribution of the external gravity field, where ρ ≃ 10³ kg m⁻³ is the water density, g ≃ 9.8 m s⁻² is the intensity of gravity and z is the altitude above ground.

• The water potential of a humid vapor, as found inside leaves where gas exchange with the atmosphere takes place, is given by:

where [RH] = P_vap/P_sat(T) is the relative humidity of the atmosphere, P_vap is the partial pressure of water in the vapor, P_sat(T) is the saturation pressure of water in air⁴ under atmospheric pressure and ambient temperature T (P_sat = 2.3 × 10³ Pa at 20 °C) and T/v_w ≈ 135 MPa at 300 K⁵.

• Finally, water is also found inside the cell wall of plants, where it binds with the cellulose network and other macromolecules such as hemicelluloses, pectins, and xyloglucans. The cell wall can be highly hydrated, as in the primary wall of growing cells, where about 75–80% of the wall volume is water,¹⁵⁵  or less hydrated, as in the lignified secondary wall of wood. The water potential of a gel-like medium akin to the cell wall is called the matrix (or matric) potential and denoted Ψ_m. The matrix potential of the cell wall depends on its water content, pH, temperature and the chemical affinity between the water molecules and the polymer constituents of the wall, mainly.¹¹²  We provide in Box 1.4 an expression of the matrix potential for an ideal elastomeric hydrogel, derived from the theory of polymer solutions.

At equilibrium, the chemical potential of water, and thus the water potential, must be uniform in all regions that can freely exchange water molecules. This basic thermodynamic statement has key consequences for the mechanical status of water in plants, which we briefly discuss below.

First consider a plant cell containing solutes of concentration c immersed in an external bath of pure water at ambient atmosphere (Ψ_bath = 0) (Figure 1.2a). We assume that the cell is surrounded by an ideal semi-permeable membrane that allows water to pass freely through it but keeps solutes inside. Using eqn (1.2), equality of the water potential between the cell and the bath (Ψ_cell = Ψ_bath) implies that there exists a positive pressure difference, or turgor pressure, between the inside of the cell and the outside, given by P = Π = cT. At equilibrium, the turgor pressure is thus equal to the osmotic pressure. For a solute concentration c = 0.2 mol L⁻¹ = 200 mol m⁻³, the turgor pressure is P = 0.5 MPa. This is twice the pressure of a car tyre and larger than the typical tension T_muscle ∼ 0.2 MPa produced by actin-myosin molecular motors and muscle-like fibers.¹³⁴  Such a large value of turgor pressure is found routinely in hydrated plant cells, and can even reach up to 4 MPa – 40 bars! – in the stomata⁵⁵  or in the shoot apical meristem of Arabidopsis.⁹³  These high positive values of the turgor pressure can be achieved in plants because the cell is surrounded by a stiff cell wall (see Box 1.1). As we shall see in this chapter and throughout the book, turgor pressure plays a key role in plant biomechanics and physiology. It determines the rigidity of non-woody plant tissues and drives irreversible cell-wall deformation during growth. Change of turgor pressure driven by osmotic or evaporation gradients is also responsible for many reversible movements in plants.^39,71 

The second application we consider is an evaporating cell in contact with a humid atmosphere at 1 bar (Figure 1.2b). If thermodynamic equilibrium is reached, equality of the water potential between the cell and the vapor (Ψ_cell = Ψ_atm) implies from eqn (1.2) and (1.3) a turgor pressure inside the cell given by P = cT + (T/v_w)ln[RH]. For a 90% humidity, i.e. [RH] = 0.9, and a solution concentration c = 0.2 mol L⁻¹ as before, the turgor pressure is P = −14.5 MPa or −145 bars! In this situation, the turgor pressure is therefore highly negative, which means that the liquid water inside the cell is in tension and pulls on the cell wall. Large negative water pressures due to evaporation are indeed observed in plant cells with very rigid cell walls, such as in the woody cells of the xylem in trees. However, liquid water in tension is in a metastable state. If a gas germ of sufficient size nucleates or is already present inside the cell, cavitation may occur: the microscopic gas germ suddenly expands into a macroscopic bubble. The mechanism of bubble cavitation and its consequence for embolism formation in trees is discussed in Chapter 4. Negative water pressure, and more generally water tension in the cell wall of dead tissues, is also responsible for many passive movements in response to humidity change,^19,76  such as the opening and closing of pine cones,³⁶  the bending of wheat awns⁴⁵  or the curling of seed pods.^4,51  These hygroscopic movements are discussed in Chapter 7.

In the aforementioned examples, the change of water pressure is induced by osmotic or humidity gradients. We conclude by considering the more common situation of a pure water column in static equilibrium under gravity, as found in the vessels of a tall tree in the absence of flow (Figure 1.2c). At equilibrium, the water potential of the sap P + ρgz given by eqn (1.2) must be uniform along the column. Assuming that the water potential is zero, which corresponds to a well-watered soil, the water pressure at the top of the tree is negative and given by P = −ρgH ≃ −0.5 MPa for a 50 meter-tall tree of height H. It is interesting to compute the size of the capillary meniscus needed to support this negative pressure difference mechanically. The Young–Laplace law states that the pressure difference ΔP_cap between water and air at a curved spherical interface is given by:

where γ ∼ 0.07 N m⁻¹ is the surface tension of the water/air interface and r is the radius of curvature of the meniscus, counted positive if the meniscus is curved toward the air. For ΔP_cap ∼ −ρgH ∼ −0.5 MPa, the radius of curvature of the meniscus is about 0.3 µm. This value is at least one order of magnitude smaller than the size of the smallest conductive vessels of the xylem found in leaves. Therefore, air/liquid capillary suction in the xylem cannot be the mechanism of sap ascent in trees. By contrast, the wall of plant cells is a hydrogel with nanometer-size pores that are small enough to support a very large mismatch of capillary pressure across the water/air interface (Figure 1.2c, close-up). Therefore, the negative water pressure in the vascular system of plants is mechanically balanced at the cell wall level, thanks to the reduction of the matrix water potential: Ψ_m = −ρgH. This equilibrium of water under negative pressure holds only if there exists a continuous path of liquid from the ground to the leaves in contact with the atmosphere, where water eventually evaporates. The coupling between water in tension inside the xylem, capillary suction by the cell wall and evaporation at the cell wall/air interface is the basis of the cohesion-theory of sap ascent.³⁸  The reader interested in xylem flows can consult Tyree and Zimmermann's book¹⁶⁰  or recent reviews such as ref. 75 and 153.

In the previous examples we have considered situations of thermodynamic equilibrium, corresponding to a uniform water potential and no water flux. Water transport requires the presence of water potential gradients, which can arise from pressure gradients, solute concentration gradients, or humidity gradients. We now discuss the laws relating these gradients to water fluxes.

Flows inside plants occur in tiny pipes. The largest conducting vessels in trees are barely larger than 0.1 mm in diameter (with a range of diameters from 10 µm to 500 µm¹⁶⁰ ), while water pores inside the cell wall or the cell membrane are nanometric in size. Hence, although the pressure difference driving water transport in plants may be very large (∼MPa range), the viscous dissipation is huge, leading to small flow velocities. For example, the typical velocity U of the sap inside the xylem of trees during the day is a few meters per hour (∼1 mm s⁻¹) for a vessel diameter d ∼ 100 µm.¹⁶⁰  The Reynolds number of this flow, which compares inertial to viscous effects, is:

where ρ ≃ 10³ kg m⁻³ is the sap density and η ≃ 10⁻³ Pa s is the sap viscosity, giving a Reynolds number Re ∼ 0.1 smaller than 1. Similarly, change of velocity usually occurs on timescales T from minutes to a day, which are much longer than the timescale of momentum diffusion across the diameter of the pipe given by τ ∼ ρd²/η ∼ 0.01 s, such that the Stokes number, defined as:

is much smaller than 1.

Therefore, inertial effects can usually be ignored when dealing with water transport in plants. In this situation, the Navier–Stokes equation of fluid mechanics reduces to the Stokes equation, balancing the pressure gradient and the viscous stress:

whereas mass conservation imposes that:

where U is the local velocity field of the flow and g the gravity vector.

Integrating eqn (1.7) and (1.8) in the case of an infinitely long, vertical pipe of circular cross-section of diameter d, like the conducting vessels of the xylem (Figure 1.2d), gives Hagen–Poiseuille's law, relating the flow rate Q (unit m³ s⁻¹) and the pressure gradients along the pipe:

One recognizes a linear relationship between the flow rate (the “flux”) and the gradient of water potential (the “force”)⁶. The coefficient of proportionality defines the conductivity K of the pipe (conductance per unit length), which varies as the vessel diameter to the power 4. Hence, when the vessel diameter decreases in an ideal branch by N identical pipes in parallel, an increasing number of vessels N ∝ d⁻⁴ is needed to achieve the same flow rate under a given pressure gradient.

The previous relationship can be generalized to pressure-driven flows in a continuum porous material (Darcy's law):

where J is the flow rate per unit area (unit m s⁻¹) and κ (unit m²) is the hydraulic permeability of the porous material (Figure 1.2d). The hydraulic permeability scales with the pore size d as κ ∝ d² and may be anisotropic, like in the xylem where flow in the longitudinal direction along the vessels is much easier than in the transverse direction. Assuming an ideal tree branch made of N identical pipes of diameter d in parallel, it is possible to explicitly find the relationship between κ and d. The flow rate per unit area is J = NQ/S, where Q is the flow rate across a single pipe and S is the area of the cross-section of the branch. Using Hagen–Poiseuille's law (eqn (1.9)) and identifying with Darcy's law gives an effective longitudinal permeability for the branch κ_xylem = (d²/32)ϕ_xylem, where ϕ_xylem = Nπ (d²/4)/S is the surface fraction of xylem. Typical order of magnitudes for d ∼ 10–100 μm and ϕ_xylem ∼ 0.2 give permeability values in the range κ_xylem ∼ 10⁻¹³–10⁻¹¹ m², similar to the permeability of fine granular soils (silt). By contrast, the hydraulic permeability of the cell wall is much lower, κ_wall ≃ 10⁻¹⁷ m², which corresponds to water pores in the cellulosic matrix of nanometric size d ∼ 3–4 nm.^20,49  For such an hydrogel made of microscopic pores, Darcy's law can be generalized as J = −(κ/η)∇Ψ_wall, with Ψ_wall = Ψ_m + ρgz (see Box 1.4).

Pressure-driven flows are very efficient in carrying water over long distances and are the main mode of water transport in the apoplast of plants (xylem, cell wall). Another important mode of transport in plants is osmosis, which occurs when solute concentration gradients exist across the semi-permeable membranes of living cells.

The transport of water and solute across a semi-permeable membrane like the plasma membrane of cells can be described using the framework of irreversible thermodynamics and the linear response between “forces” and “flux” (Onsager relations).^35,47,77,148  In the presence of a pressure difference ΔP and concentration difference Δc across the membrane, a volume flux (essentially water) J (unit m s⁻¹) and a molar flux of solute j_S (unit mol m⁻² s⁻¹) cross the membrane (Figure 1.4a), given by⁷:

On the right-hand side of the water transport eqn (1.11), the first term represents the pressure-driven flow, where L_p is the hydraulic conductivity of the membrane (unit m s⁻¹ Pa⁻¹), the second term is the osmotic flow driven by the solute concentration difference, and σ_S is the reflection coefficient of the solute (0 ≤ σ_S ≤ 1)⁸. Physically, osmosis arises because some of the momentum associated with solute thermal agitation is taken up by the cell membrane and not by the water molecules, creating a net suction of water across the membrane to the more solute-concentrated compartment (see Box 1.3 and ref. 84).

On the right-hand side of the solute transport eqn (1.12), the first term corresponds to the passive diffusion of solute along the concentration gradient (Fick's law type), where P_S (unit m s⁻¹) is the membrane permeability to solute. In plant membranes, both P_S and σ_S are solute dependent and can be regulated through the plasmodesmata permeability, in a complex feedback with turgor pressure.⁶⁸  The second term corresponds to the coupling between water transport and solute transport across the membrane, where c̄ is the mean concentration of solute across the membrane. Finally, the third term j$S*$ corresponds to the molar flux of solute actively pumped against the solute gradient using specialized channels and external chemical energy. Values ranging from j$S*$ = 10⁻⁷ to 10⁻⁶ mol m⁻² s⁻¹ are found for the potassium ion flux responsible for stomatal movement¹⁴¹  and in the motor cells of Mimosa pudica.¹⁵²  The highest solute conductance in plant membranes is probably found in MscS-like (small conductance mechanosensitive ion channel) stretch-activated channels.⁶⁵ 

In many situations (salts, sugar, metabolites) the reflection coefficient of the solute is close to unity and the membrane may be assumed perfectly impermeable to the solutes (σ_S = 1, P_S = 0). For such an ideal semipermeable membrane, the water flux across the membrane is just given by the water potential difference across the cell and the hydraulic conductivity of the membrane L_p:

Typical values of the membrane conductivity in plant cells are 10⁻¹³ < L_p < 10⁻¹¹ m s⁻¹ Pa⁻¹ depending on the physiological state of the cell.¹⁴⁷  These values actually reflect two distinct modes of water transport at the microscopic level (see Box 1.3 and Figure 1.4b). On the one hand, water can cross the cell membrane by molecular diffusion inside the lipid bilayer, which acts as a low-solubility medium for water. On the other hand, water can flow in bulk through selective water channels of high conductivity in the cell membrane, called aquaporins,⁹⁶  whose shape is optimized for water flow.⁵⁹  The opening and closing of these channels is under tight physiological control.

We conclude this brief survey by noting that the fluxes in eqn (1.11) and (1.12) are set by the value of the concentration jump determined exactly at the membrane. However, any osmotic flow – or solute diffusion if the membrane is leaky – tends to pile up solutes on one side of the membrane and sweep them away on the other side. The concentration difference across the membrane is then smaller than the concentration difference imposed in the bulk, which in turn reduces the osmotic flow. To minimize this effect, some mixing is necessary to homogenize the concentration field and bring the bulk concentration c_b to a distance δ_SL as close as possible to the membrane, called the “unstirred layer” thickness^35,120  (Figure 1.4c). In plant cells, such mixing can be achieved thanks to intracellular flows called “cytoplasmic streaming” that are actively driven by molecular motors (see Chapter 2). If δ_SL is known, it is possible to estimate the real solute concentration at the membrane c_m = c(x = 0) by balancing, in the unstrirred layer, the diffusive flux and the advection flux: Ddc/dx ∼ Jc with the boundary condition c(x = δ_SL) = c_b, where D (unit m² s⁻¹) is the diffusion coefficient for the solutes and J = L_pTc_m is the osmotic flux (assuming an ideal semi-permeable membrane and no pressure difference across the membrane). We deduce a transcendental equation for c_m:

which can be used to estimate the reduction of osmotic flux causes by unstirring layer effects. In practice, for a typical osmotic flux J ∼ 1 µm s⁻¹ (corresponding to an osmotic pressure Π ∼ 1 MPa with L_p ∼ 10⁻¹² m s⁻¹ Pa⁻¹), an unstirred layer δ_SL ∼ 10 µm (a value similar to the cell width) and a coefficient of diffusion D ∼ 10⁻¹⁰ m² s⁻¹, we find that c_m ∼ 0.9 c_b, meaning that unstrirring layer effects on osmotic flows are likely to be small for plant cells. This may, however, not be the case for solute transport if the membrane is leaky.³⁵ 

Osmotic flows are the basis for the building of the positive turgor pressure in plant cells discussed in Section 1.1.1. By adjusting their solute content, plant cells pump in and out water to maintain cell hydration in dry conditions or change their volume. One of the most important cell movements driven by osmosis in plants is the closing and opening of the stomata, the cellular valves that control the exchange of gas between the plant and the atmosphere at the surface of leaves.⁵⁵ 

Another important example of osmotically driven flow in plants is the long-distance transport of sugar (Figure 1.4d). Sugar is produced locally in leaves from photosynthesis. To ensure its transport, sugar is first loaded inside living pipe-like cells called the phloem, which are adjacent to the water-filled xylem. This high concentration of sugar in the phloem decreases its water potential and creates an osmotic flow coming from the xylem, where the water potential is larger. This inward flow creates, in turn, a positive turgor pressure at the top of the phloem and thus a pressure-driven flow that transports sugar along the phloem toward the places where it is unloaded and used for growth or storage. This mechanism was first proposed by Münch in 1930.¹⁰⁷  The velocity of the flow transporting sugar is controlled by the Münch number:

where L_p is the membrane permeability of the phloem tube, L the length of the phloem and d its diameter. For Mu ≫ 1, the viscous resistance of the phloem tube dominates. The mean flow velocity is then given by the Hagen–Poiseuille's law (see eqn (1.9)): U = Q/(πd²/4) ∼ d²/(32η)cT/L, where c is the sugar concentration. By contrast for Mu ≪ 1 the dissipation from the cell membrane dominates and the velocity U is imposed by the water flux across the membrane J = L_pcT (see eqn (1.13)) and volume conservation Uπd²/4 ∼ πLdJ, giving U ∼ (4L/d)L_pcT. The Münch number is just the ratio of these two limiting case velocities. A typical value of the Münch number for the phloem is Mu ∼ 10⁻⁴ ≪ 1 (taking L = 1 cm, the typical length of a leaf stem, d ∼ 20 µm, η ∼ 2 × 10⁻³ Pa s, L_p ∼ 10⁻¹³ m s⁻¹ Pa⁻¹ and cT ∼ 1 MPa⁷⁵ ), which gives a phloem's velocity U ∼ 0.2 mm s⁻¹ consistent with observation.

The transport of sugar along the phloem is governed by an advection-diffusion equation:

where D [unit m² s⁻¹] is the diffusion coefficient of sugar molecules in water (here no sink or source of sugar are considered along the path). The ratio of the advective to the diffusive term is quantified by the Péclet number:

Taking for the diffusion coefficient of sucrose in water D = 5 × 10⁻¹⁰ m² s⁻¹, L = 1 cm (leaf stem) and U = 0.2 mm s⁻¹ gives Pe = 4 × 10³ ≫ 1. Therefore, advection by the osmotic flow clearly dominates diffusion in the long-distance sugar transport. In-depth discussion of this fascinating mode of transport can be found in ref. 75.

We have seen that the water potential within plant tissues is generally negative, with values ranging from 0 MPa for roots immersed in a well-watered soil to few -MPa at the tops of trees or for plants living in arid or saline environments. However, the water potential of the atmosphere is much lower, with Ψ_vap = −70 MPa for a typical 60% humidity (see eqn (1.3)). Water loss by evaporation is thus a critical issue faced by all terrestrial plants, which they partly solve by covering most of their surfaces with a waxy hydrophobic layer called cuticle (Figure 1.5a). However, some gas exchange with the atmosphere must be maintained in order to capture the carbon dioxine and other gases necessary for photosynthesis and plant metabolism. This exchange occurs on the surface of leaves through specialized pores called stomata. Still, a trade-off is required as many water molecules are lost through evaporation when the stomata open (typically 100 kg of water are lost for 1 kg of glucose synthesized in plants, see ref. 112), which is achieved by fine regulation of stomatal function. A general review on leaf hydraulics, from water movements through the leaf xylem to transport in the air space between the leaf cells can be found in ref. 137. The complex interplay between biological signaling and stomata dynamics is discussed in ref. 88. Here we focus on the basic physics of vapor diffusion in relation to plant evaporation.

Evaporation at the liquid/air interface within leaves and diffusion of water through the stomata represent the final mode of water transport in plants as it moves from the soil to the atmosphere. The diffusive flux of water molecule in a humid atmosphere j_water (number of moles crossing a unit area per unit time, or mol m⁻² s⁻¹) is given by Fick's law:

where D_w = 2.4 × 10⁻⁵ m² s⁻¹ is the diffusion coefficient of the water molecule in air at 20 °C and c_w is the molar concentration of water in air (unit mol m⁻³). Fick's law can be equivalently expressed in term of the vapor pressure gradient, using the gas state equation, P_vap = c_wT, or in term of the water potential of the vapor using eqn (1.3):

However, when expressed in term of the water potential, the coefficient of proportionality between the water molecular flux and the gradient of water potential is not constant but varies exponentially as: . Hence, for a given gradient of water potential, a dry atmosphere (very negative Ψ) has a much lower “conductivity” than a humid one (Ψ≃0).

As a first example of the application of Fick's law, we consider evaporation from a surface of pure water subjected to a side wind of velocity U_wind in an atmosphere of relative humidity [RH]_atm = P_vap/P_sat(T) (Fig. 1.5b). A diffusive boundary layer of size developed above the surface, where L is the typical surface size. The molecular flux of water is thus given by:

Taking a wind velocity U_wind = 10 m s⁻¹ and a lateral size L = 10 cm (leaf size) gives a diffusive boundary layer of thickness δ ∼ 0.5 mm. For a very dry atmosphere ([RH]_atm ∼ 0), this gives a molecular flux j_water ≃ 10⁻² mol m⁻³ s⁻¹ and an evaporative mass flux J_m = M_w  j_water ≃ 2 × 10⁻⁴ kg m⁻³ s⁻¹, where M_w = 18 g mol⁻¹ is the molar mass of water. This value is surprisingly close to the maximal evaporating rate reported per surface leaf area in plants.⁷⁵  This is all the more remarkable that stomatal pores covers only 1–10% of the leaf surface, the rest being almost impermeable to evaporation.

To understand this feature, we consider an ideal leaf made of a flat impermeable sheet perforated with holes (stomata) of pore size a (Figure 1.5c). The sheet is in contact with pure water at the bottom and placed in an atmosphere of water vapor concentration c_atm above. For an isolated stomata, the size of the diffusive boundary layer is fixed by the pore size a⁹. Assuming that stomata do not interact with each other, the total evaporation rate of the leaf per unit area is thus:

where n_s is the number of stomata per unit leaf area. We find that the evaporative flux from the leaf is equal to the evaporative flux from a surface of pure water (j$waterstomata$ ∼ j$waterBL$) when n_s ∼ 1/(aδ) or ϕ_s ∼ a/δ, where ϕ_s ∼ n_sa² is the surface fraction covered by the stomata. Taking δ ∼ 1 mm and a ∼ 10 µm, we find that this condition is fulfilled with stomata covering only 1% of the leaf area (ϕ_s = 0.01). Indeed, due to their small size, stomata impose a diffusion length scale much smaller than the air boundary layer, resulting in high local evaporative gradients that compensate for their small area.

Eqn (1.27) predicts the paradoxical result that, for a sufficiently high concentration of stomata, the evaporative flux from the leaf could be greater than that from of a pure water surface. This is because we have assumed that the external vapor concentration fixing the stomatal vapor flux is c_atm. This assumption is valid only if the distance between stomata, , is much larger than the air boundary layer δ. In practice, however, s ≪ δ. In this situation, the total resistance to evaporation from the leaf, defined as R_Leaf Evap = (c_sat − c_atm)/j, is the sum of the stomatal resistance R_stomata ∼ 1/(n_saD_w) ∼ a/(ϕ_sD_w) and the resistance of the air boundary layer R_BL = δ/D_w (Figure 1.5d). We find that both resistances are equal when ϕ_s ∼ a/δ, recovering the previous condition. A detailed discussion of the role of stomata interaction on leaf vapor diffusion can be found in ref. 89.

As a second application of Fick's law, it is instructive to compare, for the same difference in water potential, the mass of water transported in bulk by a pressure gradient with that transported in vapor form by diffusion in a humidity gradient (Figure 1.5e). According to Poiseuille's law (eqn (1.9)), the mass bulk flow through a pipe of diameter d and length L is Q$mbulk$ = ρ(πd⁴/128η) × (ΔΨ/L), where ρ is the density of the liquid water, η is the water viscosity and ΔΨ = ΔP is the pressure difference across the pipe. On the other hand, transport of water vapor in the same pipe by diffusion from a vapor at saturation to an external atmosphere of humidity [RH]_atm gives, using Fick's law, a diffusive mass flux: Q$mdiffusion$ = M_w(πd²/4)(D_w/T)P_sat(T)[(1 − [RH]_atm)/ΔΨ] × (ΔΨ/L). Therefore, for the same gradient of water potential, the ratio of the bulk to diffusive transport is:

Taking d ∼ 1 µm and a water potential difference for an atmosphere of 50% humidity gives: Q$mbulk$/Q$mdiffusion$ ∼ 10⁷! This shows that, even in very small conduits, bulk transport driven by a pressure gradient in the liquid state is much more efficient than vapor diffusion driven by a humidity gradient.

This simple principle is harnessed by the plant Tillandsia, which lives in the Atacama desert in Chile, to design an asymmetrical “water valve” on the surface of its leaves¹²⁸  (Figure 1.5f). The Atacama desert is probably the driest place on Earth, with a humidity as low as 15%, which corresponds to a water potential Ψ_atm = −250 MPa. However, a few days a year, fog from the coast brings a very humid atmosphere full of tiny droplets of liquid water (Ψ_atm ∼ 0 MPa). These droplets condense on the very hydrophilic surface of Tillandsia leaves and are absorbed by the leaf cells through osmosis, due to their high concentration of solute which creates a negative water potential Ψ_leaf of about −1 MPa. Remarkably, although the difference in water potential causing water absorption during wet periods (ΔΨ_abs = |Ψ_atm − Ψ_leaf| = +1 MPa) is a hundred times smaller than the difference in water potential causing water evaporation during dry periods (ΔΨ_evap = +249 MPa), the absorption flow rate is found to be a hundred times greater.¹²⁸  This corresponds to a difference in effective conductance Q/ΔΨ between the humid and dry states a few thousands.

This strong asymmetry is explained by the existence of specialized structures at the surface of Tillandsia's leaves, called trichomes, where water exchanges takes place.¹²⁸  These trichomes consist of a row of empty dead cells covered by a very thick cell wall, which separate the leaf cells from the atmosphere (Figure 1.5f). During wet periods, the wall and dead cell cavities are completely soaked with liquid water due to their hydrophilic nature; absorption then occurs in the liquid state, through an osmotically driven bulk flow limited by the leaf cell membrane permeability. By contrast, during the dry periods, the liquid/vapor interface shifts within the trichome. Transport to the outside then occurs by diffusion in the vapor form which, as we have seen, is very inefficient.

So far in the chapter, we have considered plants as rigid and static containers, in which water and solutes are transported according to the laws of thermodynamics and fluid mechanics. However, in order to absorb water, grow or generate movement, the cells and tissues of plants must deform. This chapter discusses the relationships between stress and deformation in the plant solid body, and how this deformation is coupled with water transport. Other examples of fluid–structure interaction in plants are discussed in Chapter 2.

The solid phase of plants is essentially made by the wall material that surrounds all plant cells (Box 1.1 and Figure 1.1). Plant cells have two types of wall.¹⁵⁵  Young growing cells and the mature cells of some tissues (leaf) are surrounded by a thin primary cell wall made of stiff cross-linked cellulose microfibrils embedded in a highly hydrated matrix of polysaccharides (hemicelluloses and pectins mainly),^32,33,72  whose thickness h is typically between 0.1 µm and a few µm. When growth stops, a thicker and more rigid secondary cell wall of lignin and other woody components may be added, giving additional strength.

Under physiological conditions, the wall of living cells is stretched by the internal, osmotically induced, turgor pressure (see Section 1.1). From a mechanics perspective, a plant cell thus behaves like a pressurized thin shell (Figure 1.6a). The tensile stress σ_wall (force per unit area) stretching the wall can be estimated by modeling the cell as a spherical shell of radius R and uniform wall thickness h ≪ R under a pressure difference P between the inside and outside. The balance of force on the half-shell imposes that the external pressure force πR²P is equal and opposite to the tensile force in the wall integrated along the wall perimeter 2πRhσ_wall (Figure 1.6a), giving:

This Young–Laplace-like relationship shows that the tensile stress in the wall may be much larger than the turgor pressure due to the geometrical amplification factor R/h. Taking a typical plant cell geometry R = 10 µm, h = 1 µm and turgor pressure P = 0.5 MPa gives σ_wall = 2.5 MPa, which is much larger than the maximal tensile stress generated by animal muscle (T_muscle ∼ 0.2 MPa¹³⁴ ). We will see that this stress level is enough to significantly deform the plant cell wall and change its cell volume.

The balance of forces written here for an isolated cell can be generalized to a tissue composed of several cells. In plants, each cell wall is glued to its neighbor by an adhesive layer called the middle lamella, which prevents the cells from sliding against each other or becoming detached (the situation of invasive growth, discussed in detail in Chapter 7, is an exception). However, unlike the case of an isolated cell, the balance of forces is generally not sufficient to determine the stress distribution in all the walls of the tissue. Even when the cell geometry and the turgor pressure are known, additional knowledge of the constitutive relation of the cell wall is usually required (see Sections 1.2.2 and 1.2.4). In most situations, the complete computation of the tissue stress can only be done by means of heavy and expensive numerical simulations, using for example Finite Element Methods.¹⁵  Modeling plant solid mechanics at the tissue and organ level thus often relies on simplified models.

The first class of model, broadly called cell-based models, considers the discrete nature of cells with different degrees of approximation. Of these methods the vertex model is one of the most popular.^26,63,93  In its two-dimensional version, the tissue is modeled as a collection of polygons representing cells, in which linear segments representing the wall are in tension and connected to vertices representing the wall junction (Figure 1.6b). The global force balance implies that, at each vertex k, the (half) sum of the pressure difference force at each face adjacent to k balances the sum of the wall tension applied to k. A detailed presentation of vertex-like approaches is given in Chapter 3.

The second class of model uses continuum mechanics to define an average stress tensor σ which integrates both the wall stress and the turgor pressure at the tissue level (see Chapter 3 for a detailed discussion of the averaged stress tensor in plant tissue). In the absence of body forces, the internal force balance is then given by:

An approximation often used to describe stresses in plant tissues is based on the fact that epidermal cells are generally much stiffer than the cells of the underlying tissue, due to their small size and thick outer wall. The epidermal layer is then assumed to support most of the turgor pressure generated by the internal cells and the entire tissue is modeled as a pressurized shell^14,67,86  (Figure 1.6c). Using this tissue–tension assumption and neglecting the bending stiffness of the epidermis, the force balance in the direction perpendicular to the epidermis reduces to a generalized Young–Laplace equation:¹⁵⁷ 

Here (σ_‖,σ_⊥) are the local principal in-plane stresses in the epidermis; (R_‖,R_⊥) are the local principal radius of curvature of the organ's surface, and P represents the turgor pressure of the cells in the bulk below the epidermis. This equation must be complemented by the force balance in the plane of the shell given by (∂σ_‖/∂_s‖ = 0) and ∂σ_⊥/∂_s⊥ = 0, where (s_‖, s_⊥) is the arc-length along the principal directions. The pressurized-shell model has been widely used in the context of plant morphogenesis to determine the tissue stress distribution in organs of various shapes, such as the shoot apical meristem or in leaves^63,69,138  (Figure 1.6d). For a spherical cap of radius R like the tip of a growing shoot (R_‖ = R_⊥ = R), the tissue stress is uniform and we recover the relationship (1.29): σ_⊥ = σ_‖ = PR/2h. For a cylinder of radius R like a stem or a root (R_‖ = ∞, R_⊥ = R), eqn (1.31) predicts that the perpendicular, hoop stress is twice the longitudinal stress: σ_⊥ = 2σ_‖ = PR/h. Finally, for a saddle crease geometry like at the cusp of a bud (R_⊥ = −r < 0 and R_‖ = +R ≫ r), one of the stresses is then compressive: σ_⊥ ≈ −Pr/h while the other is in tension: σ_‖ ≈ +PR/h (Figure 1.6d). Hence, although the internal driving force – the turgor pressure – is isotropic, stress anisotropy is more the rule than the exception in plants.

The tension of the epidermal tissue predicted by the pressurized shell model is an example of residual stress, i.e. a non-zero internal stress in a body at equilibrium and in the absence of external loading. Such residual stresses are common in plants due to wall adhesion and the absence of sliding between cells. In fact, any material inhomogeneity induced by gradients of structure, elasticity, swelling rate or growth in a plant tissue is susceptible to generate residual stresses, which may or may not be resolved by tissue buckling or stress relaxation. The feedback between the tissue shape, residual stress, and the biological response (growth law, mechano-transduction) is central to plant morphogenesis and development.¹⁴ 

As the plant wall is a living material, its mechanical description must be approached with caution, especially for growing cells. However, as long as the deformations are small enough (typically less than 5–10% in living cells), and the timescales of observation not too large, the cell wall may be approximated as an elastic solid. The simplest elastic constitutive law is Hooke's law, in which deformations are reversible and proportional to the applied stresses. For uniaxial deformation, Hooke's law reduces to:

where ε_el = ΔL/L is the elastic deformation or elastic strain and E (unit Pa) is the Young's modulus of the material (Figure 1.7a). The Young's modulus is a measure of the stiffness or compliance/softness of the material: the larger E is the stiffer the material, while the lower E is the more compliant or soft it is.

As a first application of Hooke's law, we can calculate the reversible volume change ΔV induced by a change in turgor pressure ΔP in a plant cell modeled as a thin spherical shell of thickness h ≪ R (Figure 1.7b). The force balance implies σ_wall = (R/2h)P (eqn (1.29)) and Hooke's law can be written approximately as follows¹⁰: σ_wall ≈ E_wallε_el = E_wall(R − R₀)/R₀, where E_wall is the Young's modulus of the wall and R₀ the radius of the cell in its rest state when the turgor pressure P = 0. Identifying these two expressions and assuming small deformation (R − R₀) ≪ R₀ gives: ΔP ≈ (2h/R)E_wall(ΔR/R) ≈ (2h/3R)E_wall(ΔV/V). The coefficient of proportionality of this pressure/volume relationship is called the cellular bulk modulus B_cell (unit Pa):

The cellular bulk modulus determines the water storage capacity of the cells and, as we shall see, controls the timescale of cell swelling. For xylem cells under negative pressure, it also influences the dynamics of cavitation, as discussed in Chapter 4. Importantly, the cellular bulk modulus is a measure of the Young's modulus E_wall of the cell wall, up to a geometric factor ∼h/R. Measurements of B_cell in a wide variety of cells has been made using the cell pressure probe described in Box 1.2.^30,80,151  A typical value for a growing plant cell is B_cell ∼ 5 MPa. Using eqn (1.33) for a plant cell of thickness h ∼ 1 μm and radius R ∼ 10 μm gives E_wall ∼ 75 MPa. This order of magnitude is consistent with direct measurements of cell wall Young's modulus using tensile assays, which gives E_wall ∼ 50–150 MPa for onion epidermal walls¹⁶⁷  (Figure 1.7c). The Young's modulus of the wall is even larger in giant algal cells (E_wall ∼ 0.5–2 GPa¹²⁵ ) and can exceed 25 GPa for wood fibers.⁵⁷ 

Of course, the real mechanical behavior of the cell wall is more complex than the ideal Hooke's law considered here. First, the wall of living cells typically exhibits a nonlinear strain-hardening behavior, i.e. an increase of Young's modulus with strain, together with a dissipative behavior (viscoelasticity, plasticity),^{70,78,151,167}  as found in other polymeric and gels materials (Figure 1.7c). Second, the mechanical response of the cell wall is generally anisotropic, with Young's modulus being much larger in the direction parallel to the mean orientation of the stiff cellulose microfibrils than in their perpendicular direction. Therefore, even for an isotropic loading such as that imposed by the internal turgor pressure, the deformation of a cell is generally non-isotropic. This anisotropy, combined with the microscopic heterogeneity of the wall, makes the characterization of the wall by a single Young's modulus too simplistic. For example, wall stiffness deduced from indentation methods using very small probes (sub-micrometer in size) are generally much lower (of the order of few MPa^98,166 ) than those obtained from tensile micro-machines or inferred from the value of the cellular bulk modulus. This is because indentation at nanoscales primarily probes the transverse elastic properties of the wall and the polysaccharide matrix between the cellulose nanofibrils, rather than in-plane global tensile properties.¹⁶⁶ 

The cellular bulk modulus (eqn (1.33)) characterizes the elastic response of a plant cell to volume change, i.e. when the cell wall is forced to stretch further during the deformation. It is interesting to study a situation of cell deformation without change of stretch, such as a single turgid cell squeezed between two plates by a displacement δ. When the internal turgor pressure P is high and the cell wall thickness is thin (h ≪ δ), the problem is similar to that of a squeezed elastic balloon (Figure 1.7d). The force then mainly comes from the internal pressure acting against the plates: F ∼ Pπa², where a is the radius of the contact area between the cell and the plates. Assuming that δ is small compared to the radius of the cell R gives the geometrical relation a² ∼ 2Rδ and thus a linear relationship between the force and the displacement: F ∼ 2πRPδ. From this result, one can define an effective external stress acting on the cell σ = F/(πR²), an effective elastic strain: ε_el = δ/R and thus an apparent Young's modulus of the cell E_cell = σ/ε_el given by:

Interestingly, this apparent Young's modulus depends on the turgor pressure but not on the cell wall elasticity, a property that can be used to infer the turgor pressure in experiments on single cells^41,91  (see Box 1.2). When the turgor pressure drops to zero, the apparent cell Young's modulus predicted by eqn (1.34) vanishes. In reality, there is always a small elastic resistance coming from the bending of the cell wall, which we have ignored previously. The force needed to bend a plate of size R and thickness h over a displacement δ is F ∼ E_wall(h³/R²)δ,⁸⁷  yielding an apparent bending cell modulus E_bending ∼[F/(R²)]/(δ/R) given by:

The previous analysis at the cellular level is helpful to understand the relationship between the macroscopic Young's modulus of a multicellular turgid plant tissue and the microscopic properties of its constituents (Young's modulus of the cell wall, cell geometry). This link is, however, not straightforward. Even for an idealized tissue composed of identical cells of internal turgor P, wall thickness h and size R, dimensional analysis states that the tissue Young's modulus should depend on both the turgor pressure and the cell properties through two independent dimensionless numbers, namely P/E_wall and h/R. It is possible to simplify the prediction and derive scaling relations when the turgor pressure is higher than the bending stiffness (P ≫ E_bending) so that the cell walls are all pre-stretched and in tension. As the liquid is incompressible, it is generally not possible to deform such a tissue without further stretching the walls, unlike the case of an isolated cell (Figure 1.7e). The tissue's Young's modulus E_tissue is then dominated by the stretching mode of deformation, as in the case of the cellular bulk modulus, which gives:

where ϕ ∼ h/R is the solid (wall) volume fraction of the tissue.

This scaling law is compatible with values of Young's modulus of turgid tissues E_tissue ∼10 MPa and is valid as long as E_bending ≪ P ≪ B_cell. When the turgor pressure drops to zero, for e.g. by plasmolyzing the cells in an osmotic solution, the tissue modulus sharply drops because the cell walls are no longer stretched and can bend easily. In real plant tissue, the situation is complicated by the fact that the cell wall's modulus E_wall itself is not constant but strongly varies with the deformation (Figure 1.7c), yielding complex turgor pressure dependences. Note that in all this discussion, we have assumed that the tissue remains always close to full hydration, even when the turgor drops to zero. When the tissue further loses water below the onset of plasmolysis, as during drying in air, the tissue Young's modulus E_tissue may rise again because the solid fraction ϕ increases, until it reaches the wall Young's modulus E_wall (see the discussion in Chapter 5 in the context of roots). More details on this question from the standpoint of cellular materials can be found in ref. 57, 111 and 163.

So far, we have discussed the elastic behavior of a plant cell and tissue without considering the flow of water within it. However, we have seen that the cell wall and plasma membrane are permeable to water. Therefore, any deformation resulting in a local change in turgor pressure, such as the bending of a stem (Chapter 2) or the squeezing of a fruit, must induce a movement of water within the medium. The same is true for a plant cell or tissue immersed in a solution of different osmolarity. To restore the water potential balance, the tissue must absorb or expel water and therefore must swell or shrink elastically. This coupling between flow and deformation in elastic porous media like plant tissues is described by poroelasticity or poromechanics theory.³⁴  As we shall see, poroelasticy sets the timescale of response of plant tissues to sudden change of water potential, and thus provides a bound for all water-driven movements.^39,50,146 

We first consider poroelasticity at the cellular level. Consider an isolated plant cell of initial volume V_i immersed in a bath at thermodynamic equilibrium (Ψ_bath = Ψ_cell,0 = P₀ – c₀T), where P₀ is the initial turgor pressure of the cell and c₀ is its initial solute concentration (Figure 1.8a). The water potential balance is suddenly perturbed at time t = 0, for example by suddenly changing the internal turgor pressure (a similar result holds for a perturbation of solute concentration or external water potential). What is the swelling/shrinking dynamics of the cell in response to this perturbation?

To answer this question, we use mass conservation and state that the change in volume V of the cell per unit time is equal to the flow rate of water through its surface S. Using eqn (1.13) of the water flux J across an ideal semi-permeable membrane, this gives:

where L_p is the hydraulic conductivity of the membrane. For small perturbations around the initial state, we can write: V = V₀ + δV(t), S = S₀ + δS(t), P = P₀ + δP(t) and c = c₀ + δc(t). Since inertia is negligible, the mechanical equilibrium is always satisfied and we can use the pressure/volume relationship δP = B_cell(δV/V₀) seen in Section 2.2 (eqn (1.33)). Finally, assuming the conservation of solute during the dynamics implies: δc/c₀ = −δV/V₀. At the lowest order, eqn (1.37) then reduces to a linear relaxation equation for the turgor pressure:

where τ_cell is the cell relaxation time and Π = c₀T is the osmotic pressure. Generally, B_cell ≫ Π; writing V/S ∼ R where R is the typical size of the cell thus gives:

The cell relaxation time thus increases when the cell size increases or when the membrane permeability or cell elasticity decreases. Typical measurements in giant algal cells of R = 200 µm, B_cell = 30 MPa, and L_p = 2 × 10⁻¹² ms⁻¹ Pa give τ_cell = 3.5 s.

The cell relaxation time sets the shortest response time of a plant cell to a sudden (small) change of water potential: whatever the timescale of other biochemical or genetic processes, e.g. ion flux, aquaporin opening, change of wall mechanical properties, a cell cannot change its volume in a time τ < τ_cell. The cell relaxation time thus defines the fastest possible swelling/shrinking movement at the cellular level.^39,50  This hydraulic limit is probably reached in some carnivorous fungi (e.g. Arthrobotrys brochopaga), which have evolved special mycelial structures to capture nematodes¹¹⁵  (Figure 1.8b). In these organisms, the trap consists of a constrictive ring composed of three connected cells of size R ∼ 5 μm. When a nematode enters the ring, the cells rapidly inflate inward, wrapping the nematode in less than 1/10 of second. The current hypothesis for this rapid movement is that the cell wall of the ring is bi-composite, with a very rigid outer wall surrounding a more flexible inner wall. Upon stimulation, the outer wall suddenly breaks, which is equivalent to a sudden drop in the Young's modulus of the wall. To satisfy mechanical equilibrium and Hooke's law, [σ =(R/2h)P = E_wε_el, see Section 2.2], the turgor must therefore drop abruptly, resulting in water being sucked in and the cell swelling very rapidly due to its small size (τ_cell ∝ R).

We can extend the previous approach to the tissue level. The simplest model of a plant tissue is a one-dimensional chain of identical rectangular cells in which water is transported from cell to cell through the cell plasma membranes, without considering the cell wall between the cells (Philip's model,¹²²  see Figure 1.8c). Using the water transport law (eqn (1.13)), the mass conservation for the volume V_i of the cell i is given by (assuming Ψ_bath = 0):

where A is the cell area in the direction of the flow, L$p′$ = L_p/2 is the effective membrane hydraulic conductivity (the factor 1/2 comes from the fact that two cell membranes has to be crossed per cell), P_i is the turgor pressure of cell i and Π_i = c_iT is the osmotic pressure of the cell i. Using again the small perturbation approximation and the initial water potential equilibrium condition, eqn (1.40) reduces to:

where τ_cell = V₀/[AL$p′$(B_cell + Π)] is the single cell relaxation time. Using the continuum approximation : δP_i(t) → δP(x,t), the set of discrete differential eqn (1.41) becomes a single partial differential equation:

with _Philip = R²/τ_cell ∼ RL_pB_cell.

Eqn (1.42) corresponds to a diffusion equation for the relaxation of the turgor pressure, with a diffusion coefficient _Philip. The characteristic time for water transport through a tissue of length L, called the poroelastic time τ_p, is given by:

The poroelastic time τ_p depends on the cellular hydraulic and mechanical properties and scales with the tissue size as ∝L².

The Philip's model has been extended to take into account both the cell-to-cell pathway and the apoplast pathway, i.e. through the cell wall (see Box 1.1) by Molz and Ikenberry¹⁰⁰  (Figure 1.8d). This model assumes that water can be transported from cell to cell by a gradient of cell water potential Ψ_cell, and into the wall by a gradient of hydrostatic pressure P_wall. Water can also be exchanged perpendicularly between the wall and the cells. In this framework, the water transport is described by two diffusion equations with a coupling term:

where ₁ = _Philip and ₂ = 2B_wallk_wall/2η is a water diffusion coefficient in the cell wall, with B_wall the wall bulk modulus, k_w the cell wall (Darcy) permeability and η the water viscosity. The coupling terms involve the properties of the cell wall and cell membrane: and , where ϕ ≪ 1 is the wall volume fraction.

In practice, water diffuses much faster through the wall than from cell to cell, because the wall is much more rigid than the cell (₂ ≈ 100 ₁). However, the small solid fraction of the wall and the large exchange surface between wall and cells make the coupling between the two paths very strong. In this situation, it can be shown that there is a local quasi-equilibrium between the water potential of the wall and the water potential of the adjacent cell, such that Ψ_cell ≈ P_wall = Ψ_tissue, where Ψ_tissue is the water potential of the tissue. The two coupled equations can then be reduced into a single effective diffusion equation for water transport:

where _tissue is an effective water diffusion coefficient given by:

Typical values for plant tissues (taking R ∼ 10/100 μm, L_p ∼ 10⁻¹² ms⁻¹ Pa⁻¹, B_cell ∼ 10 MPa, ϕ ∼ 0.1, B_wall ∼ 100 MPa, k_wall ∼ 10⁻¹⁸ m²) gives _tissue ∼10⁻⁹–10⁻¹⁰ m² s⁻¹, in agreement with measurements.^100,148  A three-dimensional extension of this model, taking into account the transport in vascularized tissues, can be found in ref. 130 and 131.

When the effective medium approximation is valid, one can approach water diffusion in plant tissues using continuum equations and the theory of poroelasticy (see for e.g ref. 34). We derive here a simple one-dimensional poroelastic model for a porous material made of an incompressible solid matrix of volume fraction ϕ = δV_S/δV, where δV_S is the local volume of solid and δV the local total volume, infiltrated by an incompressible liquid of volume fraction ϕ_f = δV_f/δV = 1 − ϕ, where δV_f is the local volume of fluid (Figure 1.8e). We note (resp. ) the volume averaged velocity of the solid matrix (resp. fluid). The conservation of mass written for the total medium (solid and fluid) and for the solid phase gives two equations:

Neglecting fluid and solid's inertia, the conservation of momentum for the total medium and for the fluid phase gives:

Here, σ_S is the volume-averaged stress in the solid matrix corrected by a fluid pressure term: , where P is the pressure in the fluid phase (minus the atmospheric pressure by convention). This definition of the solid stress, also called Terzaghi effective stress, ensures that the solid stress is zero when the matrix is immersed in a fluid of uniform pressure without deformation. The second equation for the fluid's momentum balance is Darcy's law, where J is the volume flux of fluid, k is the Darcy permeability of the porous medium and η is the fluid's viscosity.

The system is closed by adding a constitutive law for the mechanical behavior of the solid phase. Assuming an elastic linear relationship between stress and deformation (Hooke's law), we can write:

where ϕ₀ is the solid fraction of the undeformed matrix, (ϕ₀ − ϕ)/ϕ₀ is the one-dimensional elastic strain and E = −ϕ(dσ_S/dϕ) is a one-dimensional Young's modulus of the solid matrix.

Linearizing eqn (1.47)–(1.49) around the rest state P = 0, U_S = U_f  =  0 and ϕ = ϕ₀ gives:

One recovers a diffusion equation for the transport of water in the poroelastic medium, as obtained previously using a discrete description of the plant tissue. The effective diffusion coefficient  is a function of the fluid's viscosity η, the Darcy permeability k and the elastic Young's modulus E of the porous matrix. The poroelastic time of water diffusion across a tissue of size L is given by:

As for the cell relaxation time at the cell scale, the poroelastic time τ_p provides a bound for the fastest possible water-driven movements in plants at the tissue and organ level.¹⁴⁶  A plant motion occurring on a timescale τ > τ_p can rely on water transport, whereas systems with τ < τ_p must use other mechanisms (Figure 1.8f). The strong size dependence of the poroelastic time (τ_p ∝ L²) shows that hydraulic movements are increasingly less efficient in terms of speed as the system size increases. We shall see in Section 2.5 how the use of mechanical instabilities enables plants to overcome this hydraulic limit.

So far, we have dealt with the small deformation regime of plant tissues, where strains are mainly reversible. We now address the opposite extreme deformation regime corresponding to plant growth. Growth is the irreversible and sustained expansion of cells under physiological conditions.¹⁵⁵  It is obviously a very complex phenomenon that requires a continuous interplay between biochemistry, mechanics and hydraulics to maintain homeostasis. During growth, the cell wall must be continuously synthesized while it expands in order to maintain its mechanical and structural integrity. At the same time, the absorption of water from the surrounding environment must be precisely balanced, in order to allow the increase in cell volume and thus the expansion.

The concept of growth usually encompasses cell division, cell differentiation and the actual growth of cells, i.e. their increase in size. Cell division occurs in localized regions called meristems. In plants, a distinction is made between primary growth, responsible for the elongation of organs, and secondary growth, responsible for the radial thickening of organs. Primary growth is driven by meristems located at the tip (apex) of stems and roots, and therefore called apical meristems. The meristem of secondary growth is the cambium, a ring of cells around the axis of stems and roots. We focus here mainly on primary growth and on the mechanical aspects of cell expansion, leaving aside the question of cell division and differentiation. A discussion of growth from a modeling perspective is found in Chapter 4.

The basic mechanics of plant growth have been revealed by seminal experiments conducted on giant algal cells.^60,125–127  First, growth requires the existence of a minimum turgor pressure to occur, i.e. a minimum stretch or stress for the cell wall to yield. Second, above this yield pressure, the rate of cell expansion, or growth rate, depends on the pressure difference between the turgor pressure and the yield pressure. Third, when applying increments of turgor pressure, the growth rate stabilizes at its steady value only after a given relaxation time.

These observations are reminiscent of the behavior of yield-stress fluids in soft matter, such as emulsions, polymeric gels or clays, which exhibit elastic behavior below a yield-stress and viscous behavior above.⁶  From the mechanical standpoint, the simplest constitutive law of the plant cell wall is therefore that of a Bingham fluid or an elasto-viscoplastic fluid. Such Bingham model was first proposed by Lockhart in 1965⁹²  and then extended by Ortega in 1985¹¹⁹  to take into account elasticity. For uniaxial deformation, the Lockhart–Ortega rheology of the cell wall can be written as:

In this model, the total strain in the cell wall ε is decomposed into an elastic part, ε_el, and an irreversible or plastic part, ε_ir, which represents growth (Figure 1.9a). Below a critical stress σ_Y, the cell wall behaves as an elastic solid (ε = ε_el) and follows Hooke's law with a Young's modulus E_wall. Above σ_Y, the cell wall deforms irreversibly and flows with a strain rate _ir proportional to the excess stress (σ_wall − σ_Y). The proportionality coefficient Φ_wall between the strain rate and the excess stress is called the extensibility of the cell wall and is associated with the irreversible creep of the wall under stress, also called wall-loosening.¹⁶⁶  Dimensionally, the cell wall extensibility is an inverse viscosity [unit (Pa s)⁻¹]. However, it should be kept in mind that the extensibility reflects a complex process at the microscopic level, including the addition of mass and a chemically mediated remodeling of wall material. The extensibility strongly depends on the temperature – a sign that it is under the control of cell metabolism¹²⁷  – and can be modified under the action of pH-dependent wall-loosening enzymes, such as expansins.^25,32  Several molecular-scale^2,42,132,167  or thermodynamic⁷  models have been proposed to describe this active process, some of which recovering a Bingham-like rheology at the macroscopic level (see Chapter 4 for a detailed presentation of one of these microscopic models).

The simple uniaxial Lockhart–Ortega model (1.60) has been extended to anisotropic cell wall under multiaxial stress, either by assuming an anisotropic extensibility⁴⁰  or an anisotropic elasticity by analogy with fiber-reinforced composite materials.^15,18  A more complex constitutive law explicitly accounting for the reorientation of microfibrils under flow has been also developed, by analogy with liquid crystals.⁴³  For such tensorial laws, the question of whether the cell wall yields above a critical strain ε_Y or a critical stress σ_Y, which is equivalent in the one-dimensional Lockhart model, is of importance, as the principal direction of strain may not coincide with the principal direction of stress.¹¹⁷  For elongated cells and organs for which σ_⊥ > σ_‖ (see Section 1.2.1), the direction of growth is mainly longitudinal and thus perpendicular to the maximal stress, suggesting that the yield condition in terms of strain is more relevant.

Finally, a growing body of evidence suggests that the cell wall rheology is regulated not only by molecular signals, but also by the mechanical state of the wall itself.¹⁵⁹  A key molecular actor of this feedback is the cortical microtubules, whose orientation determines the orientation of the cellulose microfibrils in the wall, and thus wall anisotropy (see Box 1.1. and Figure 1.1b). Experiments show that the orientation of the microtubules aligns with the direction of maximal stress in the wall, generating a feedback loop between the organ's shape (determining stress direction) and material anisotropy (determining growth direction and thus shape)⁶³  (Figure 1.9b). This mechanical feedback is taken into account in the most recent rheological models¹¹⁸  but the precise transduction mechanism is still debated.^64,117  More generally, how mechanics modified growth, either through direct effect on the wall stress or through the perception and transduction of internal and external mechanical signals – a process known as thigmomorphogenesis – is a very active area of research in plant biophysics. A discussion of these aspects in the context of root growth in given in Chapter 5.

The Lockhart–Ortega rheology can be used to describe the unidirectional and uniform steady growth of an isolated cell, modeled as a cylinder of fixed radius R and length L(t), immersed in an external bath of water potential Ψ_bath (Figure 1.9c). This situation typically corresponds to the growth of the giant internode cells of the green algae Chara or Nitella. For these cells, the wall expansion is evenly distributed over the wall surface and growth is uniform along the cell, a mode of growth called diffuse growth¹¹. Moreover, the cellulose microfibrils in these cells run mainly circumferentially, restricting growth in the radial direction and promoting growth in length.

In this cylindrical geometry, the balance between the turgor pressure and the wall stress implies that the longitudinal wall stress is given by σ_wall = PR/(2h), where h is the wall thickness and P the turgor pressure. For a steady state of growth the elastic strain ε_el is constant. The growth strain rate, also termed relative elongation rate (RER) or relative elementary growth rate (REGR) in the plant science community, is then given by  = _ir = (1/L)(dL/dt). Combining the force balance and the Lockhart–Ortega wall rheology (1.60) thus implies:

where Φ = (Φ_wallR/2h) is an effective extensibility defined at the cellular level and P_Y = 2hσ_Y/R is the yield turgor pressure for growth. This first Lockhart's equation must be complemented by a second equation for water transport. Indeed, for cell expansion to be possible, an inward flow of water must compensate for the increase in volume, which requires a difference in water potential between the inside and outside of the cell. Using the water transport eqn (1.13), the second Lockhart equation is:

where L_p is the hydraulic conductivity of the membrane and c the cell solute concentration. To close the system, a third Lockhart equation should be added to describe the temporal evolution of the solute concentration.¹⁰⁵  We assume here that growth is slow enough that osmotic balance is maintained, so that solute concentration remains constant. The solution of (1.61–1.62) is therefore:

Two limiting regimes can be considered depending on the value of the Lockhart's parameter: ℒ = 2L_p/RΦ. For ℒ ≫ 1 (small extensibility, large membrane permeability), eqn (1.63) becomes:

This regime is called the extensibility-limited regime. In this case the growth rate of the cell is set by how fast the cell can extend its cell wall (embedded in the parameter Φ), not by its ability to take up water across the plasma membrane. The water potential balance is then almost satisfied and the turgor pressure is simply given by the osmotic pressure cT. Growth of most plant organs is assumed to operate in this extensibility-limited regime. From typical growth rate in young shoots  ∼ 0.1 − 1 h⁻¹ (see ref. 144) and using standard values cT ∼ 0.5 MPa, P_Y ∼ 0.3 MPa³¹  and R/2h ∼ 10, we find that the cell wall “viscosity” is 1/Φ_wall ∼ 10–100 GPas – a value close to the viscosity of ice!⁵⁴ 

For the opposite regime ℒ ≪ 1 (large extensibility, small membrane permeability), the Lockhart eqn (1.63) becomes:

In this conductivity-limited regime, the extensibility of the cell is so large that the growth rate is limited, not by the cell-wall extensibility, but by the water uptake resistance set by the plasma membrane. The turgor pressure is then poised at the yield pressure P_Y, not at the osmotic pressure , meaning that a significant water potential gradient exists between the inside and outside of the cell.

Since the pioneering work of Molz and Boyer,⁹⁹  the Lockhart equations have been extended to multicellular tissues using cell-based models to couple water transport and growth.²⁶  A scaling analysis of the problem can also be made using the same kind of continuum approach as we used for poroelasticty (Section 2.3). Consider a tissue of size L growing at a constant rate . Volume conservation requires a water flux to sustain growth, which is given by: J = (1/S)dV/dt = L, where V is the volume of the growing tissue and S the cross-section perpendicular to the flow. From Darcy's law, this water flux must be induced by a turgor pressure gradient across the tissue (assuming uniform solute concentration) given by: (ΔP/L) = (η/k)J ∼ (η/k)L, such that:

This relationship is the tissue version of the second Lockhart equation (1.62) on water transport. On the other hand, the force balance between the internal turgor pressure and the cell wall still holds at the tissue level, such that σ ∼ (R/h)P. Using the Lockhart–Ortega rheology for the wall in steady state, we recover the same relationship as the first Lockhart's equation:

This analysis shows that the gradient of turgor pressure induced by tissue growth is negligible if ΔP ≪ P − P_Y, which using eqn (1.66) gives:

The dimensionless number ℒ_tissue is the tissue equivalent of the Lockhart number ℒ defined at the cellular level. For ℒ_tissue ≫ 1, the tissue growth rate is limited by cell-wall extensibility while for ℒ_tissue ≪ 1, it is limited by water conductivity. Interestingly, this “tissue Lockhart number” depends not only on the mechanics and hydraulic properties of the tissue but also on the system size, with a strong size dependence ℒ_tissue∝L⁻². It has long been assumed that growth in plants operates in the extensibility-limited regime and that growth-associated water potential gradients were small, with the exception of fast-growing tissues such as coleoptiles or roots.^{16,17,30,31,99,148}  However, recent measurements of turgor pressure at cellular resolution in the shoot apical meristem of Arabidopsis revealed high cell-to-cell pressure heterogeneity, which can only be predicted by taken into account water transport in addition to wall mechanics.⁹³  Water conductivity was also shown to strongly influence the emergence of lateral roots in Arabidopsis.¹²¹  Therefore, the role of water conductivity in the control of growth and development might be more important than previously anticipated, even at small scales.

In the previous section, we considered uniform and unidirectional growth. When growth varies spatially, kinematic constraints may lead to a change in growth direction and thus in organ shape. We consider here the important case of differential growth in a slender, rod-like body, such as a plant shoot or stem, which leads to the bending of the organ.^9,105,144 

To address this question, we model the shoot by a thin rod of length L(t) and constant radius R ≪ L, restricting its shape to a curve in two dimensions for simplicity. The spatio-temporal shape of the rod is then fully described by the angle θ(s,t) made by the rod with the vertical axis, where s is the arc length from the fixed base and t is time (Figure 1.10a). By definition, the local curvature of the rod is: C(s,t) = −∂θ/∂s. Here the convention is such that curvature is positive when the shoot angle decreases away from the base, as in Figure 1.10a. The growth kinematics are characterized by the velocity field v(s, t) of the different material points along the rod. A portion ds of the rod at position s and time t increases to a length (1 +  dt)ds at time t + dt, where  = ∂v/∂s is the local growth rate. Similarly, the Lagrangian change of curvature during dt, i.e. following the material displacement, is:

We now assume that the growth rate not only varies along the rod but also across the thickness of the rod, from a value ₂ at the bottom side of the rod to a value ₁ < ₂ at the upper side (Figure 1.10a). During dt, the lower side length thus expands to (1 + ₂dt)ds while the upper side expands to (1 + ₁dt)ds. As shown in Figure 1.10a, this differential expansion must induce a small change of rod curvature dC. Geometry imposes that: [(1 + ₂dt)ds]/(r + R) = [(1 + ₁dt)ds]/(r − R), where r = (1/dC) ≫ R. At first order, the curvature change induced by differential growth is then given by:

Combining this kinematic growth relation with the expression (1.69) of the Lagrangian derivative gives:

This equation shows that change of curvature in a thin elongated organ is driven by differential growth, here expressed in terms of the relative growth asymmetry Δ = (₂ − ₁)/(₂ + ₁). For a given Δ, the rate of change of curvature is set by the mean growth rate  = (₂ + ₁)/2 = ∂v(s,t)∂s. Note that local change in curvature impacts the entire organ orientation because the relationship between angle and curvature is non-local:

where θ₀ is the angle of the shoot at the base.

Eqn (1.71) shows that maintaining a steady curved shape in a growing organ (∂C/∂t = 0) requires a subtle spatio-temporal regulation of differential growth to satisfy: v(s,t)R(∂C/∂s) = (s,t)Δ(s,t) – an idea first put forward by Wendy Silk (see ref. 145). Such steady growing shapes are found in many seedlings¹⁴⁵  and also in some compound leaves.¹²⁹  In these examples, the end of the shoot or stem exhibits a hook shape that remains steady and at a fixed distance from the plant apex, all along the growth. Although the hook shape appears steady, each tissue element must follow a complex bending and unbending growth pattern as it moves from the tip, in order to comply with kinematics (see Figure 1.10b).

Bending induced by differential growth is also the basis of most tropic movements in plants,⁵⁸  i.e. the directed motion in response to external stimuli such as light (phototropism), touch (thigmotropism) or gravity (gravitropism), which we discuss below. Plants detect gravity using specialized cells (called statocytes), in which starch-rich particles (called amyloplasts or statoliths) sediment under gravity to form miniature “granular” piles at the bottom of the cell^{101,103,109,158}  (see Figure 1.11 and Box 1.5). When the cell is inclined, the statoliths move toward the side of the cell even at a small inclination, unlike sand grains, because statoliths are fluidized by the cytoskeleton's activity (see ref. 13 and Box 1.5). The resulting asymmetric position of statoliths within the cell triggers a complex molecular pathway, leading to the generation of a growth-hormone gradient across the thickness (the growth hormone in plants is called auxin) and thus to differential growth and organ curvature.

The gravitropic response of plant shoots is inclination-dependent and follows a sine-like shape, such that:

where β is a dimensionless parameter characterizing the gravitropic sensitivity of the plant. This sine-law has classically been interpreted as the consequence of a force-sensing mechanism at the cellular level, the projection of the weight of the statoliths along the side of the cell being proportional to sin θ. However, recent experiments have shown that the response is independent of the gravity intensity¹², suggesting that the gravitropic stimulus is the position of the statoliths within the cell, not their weight.^24,124 

Surprisingly, when the expression (1.74) of the gravitropic response is introduced in eqn (1.71)–(1.72) (without the advective term) to predict the gravitropic bending of an inclined stem, an unrealistic spatio-temporal dynamics is found. Instead of converging toward the vertical, the tip of the stem keeps oscillating back and forth, with regions of increasing curvature accumulating close to the base of the stem.⁸  This paradox was solved by adding to the gravitropic law (1.74) an additional term proportional to the local curvature, which counteracts the generation of curvature induced by gravity sensing:⁸ 

where γ is a dimensionless parameter. Such tendency of a stem or a shoot to perceive its own curvature and straighten was coined “autotropism” in early studies (see ref. 21 and 48 and older references therein) and latterly “proprioception”, by analogy with the sense of body perception in vertebrates.⁸  In plants, the mechanism of proprioception is still debated^104,116  but its reality is well supported, in particular by experiments in microgravity environments which show that an initially curved stem spontaneously straightens during growth. Writing for simplicity sin θ ≈ θ and neglecting the advection term in eqn (1.71) gives for the evolution of the curvature:

This mathematical model of gravitropism, which includes both graviperception and proprioception, was first introduced by Bastien et al.⁸  and coined the “AC model” (for Angle–Curvature). Unlike the purely gravitropic model, the AC model exhibits a steady solution (∂_tC = 0) given by θ(s) = θ₀exp(−s/L_c), where θ₀ is the inclination of the stem at the base and L_c= γR/β is a bending length scale. This steady shape is controlled by the balance number ℬ:

which quantifies the ratio of the gravitropic term to the proprioceptive term. For ℬ ≫ 1 (L_c ≪ L), graviperception is large compare to proprioception and the plant rapidly recovers the vertical after a bending length L_c. In the opposite situation ℬ ≪ 1 (L_c ≫ L), proprioception overcomes graviperception and the plant does not recover the vertical over its length L. Note that the balance number ℬ not only controls the final shape of the plant but also its dynamics toward vertical: for large ℬ, the tip of the stem oscillates several times around the vertical before converging, which is not the case for small ℬ.⁸ 

The previous discussion shows that the perception of gravity alone through the sine law (1.74) does not enable a proper posture control in plants, and that an additional straightening or proprioceptive mechanism is needed. The AC model constitutes the building block from which additional effects may be considered. For instance, the gravitropic response exhibits several timescales not included in the simple sine law, such as a delay time before the beginning of bending and a memory time that filters rapid changes in inclination.²³  The purely kinematic model presented here can also be extended to add mechanical effects, such as elasticity and stem sagging under its own weight.²⁷  In this case, the actual (or observed) curvature C contains an elastic contribution due to the self-weight. It must thus be distinguished from the natural¹³ curvature C⁰ induced by differential growth and described by eqn (1.71). Finally, other kinds of tropisms and plant movements may be included in this framework, such as phototropism, thigmotropism or circumnutation – the spontaneous oscillation and circular movements of the tip of stems.^1,10,106  Extension to three-dimensional growing rods, including torsion and helicity, are also available (see ref. 106 and references therein).

We have seen how differential growth across the thickness of a rod-shaped organ like a stem induces a change in natural curvature. When the organ remains thin but extends in two directions, such as in a flat leaf, two different types of differential growth-induced shapes can occur¹⁴.

The first mode of deformation is similar to the one we just saw in the case of a one-dimensional stem and comes from a differential growth across the thickness of the sheet. In this situation, a natural curvature¹⁵ C⁰ in the direction of differential growth is generated, causing the sheet to curl into a cylindrical shape. Many plant leaves, for example grass blades or corn leaves, exhibit such rolled shapes induced by a differential expansion across the leaf thickness. However, when the leaf is curved in two perpendicular directions like a shell, bending in one direction is coupled to the other direction through the generation of in-plane tissue stresses, causing subtle rolling/unrolling mechanisms.¹⁰² 

The second type of deformation is associated with differential surface growth: different regions of the sheet grow at different rates (Figure 1.10c). In general, these spatial variations in growth cannot happen without generating in-plane compressive stresses. For a thin sheet of lateral size L and thickness h ≪ L, the energy cost of such compressive deformation is very high and the system prefers to bulge out of the plane. A non-zero natural Gauss curvature κ$G0$ is then created¹⁶. When the growth is larger in the center of the sheet than at the edges, the generated Gauss curvature is positive κ$G0$ > 0, i.e. the sheet deforms into a dome shape. In contrast, when the edges grow faster than the center, negative Gauss curvature κ$G0$ < 0 is created, resulting in rippling edges and saddle-like shapes. Many flat organs in plants and algae, such as leaves, blades or flowers, show this type of rippling at their edges, induced by faster growth at their periphery.^{29,83,142,143} 

Overall, geometry and mechanics, in addition to genetics, play a primary role in morphogenesis and development – an idea promoted by pioneering scientists such as d'Arcy Thompson¹⁵⁶  and Paul Green.⁶¹  This role can be purely “passive”, as when differential growth generates out-of-plane bending to satisfy geometric compatibility. It can also be “active” and directly influence the molecular and genetic machinery of the plant, as we have seen in the case of the cell wall, where the stress in the wall controls the orientation of the cortical microtubules, which in turn affects wall anisotropy⁶³  (Figure 1.9b). Understanding such feedback between mechanical signals and biology is one of the main current challenges in plant morphogenesis.¹³⁹ 

Throughout the chapter we have seen how plants, although lacking muscle, use gradients of water potential to transport water and perform various swelling or growth movements. The timescale of these water-driven movements is, however, constrained by a physical limit: for a cell or a tissue to swell, water must be transported from one place to another within the soft, porous plant material. We have seen that the shortest response time of this process is set by the cell relaxation time at the cellular level, and by the poroelastic time at the tissue level (Section 1.2.3). These timescales depend on the hydraulic and elastic properties of the medium and, more importantly, on the size of the system: the larger the organ, the greater the time required to produce purely hydraulic motion.

Plants have developed a simple and elegant strategy to overcome this hydraulic limit: the use of mechanical instabilities.^50,52  The general principle is as follows: during a first “slow” phase, elastic deformation is stored in the cell wall due to a slow modification of the hydraulic or elastic properties (e.g. a water movement due to evaporation or active solute transport, a change of the mechanical properties of the wall, etc). However, this elastic energy is not released because there exists an “energy barrier” in the system. When the stored elastic energy reaches a critical point, the energy barrier is crossed and a second “fast” phase occurs where the deformation in the wall is rapidly released and converted into a fast movement.

The existence of an energy barrier is the key ingredient for developing a mechanical instability. This barrier can be of molecular origin, as for bubble nucleation in a liquid under negative pressure or crack nucleation in a solid under tension. In this case, the cohesion between molecules is associated with an energy cost to create a new surface – the bubble or crack – in the medium. When the size of the bubble (or length of the crack) is large enough, the release of bulk energy is larger than the cost of surface energy; the bubble (or crack) then expands suddenly. Fracture propagation and its associated explosive elastic energy release is used by many plants to disperse their seeds or spores.^{37,46,74,76,133}  The cavitation of bubbles in a liquid under tension, so harmful for trees, has been harnessed by some species of ferns to catapult their spores and thereby generate the fastest movement in the plant kingdom.^81,113  The rich physics of cavitation in the context of plants is discussed in Chapter 4.

Mechanisms based on fracture propagation or bubble cavitation are one-shot, irreversible fast movements: once the stretched tissue is torn or the metastable liquid transformed into vapor, no resetting of motion is possible. A more flexible strategy to actuate reversible rapid movements consists of using an elastic or snap-buckling instability. A snap-buckling instability is the discontinuous transition between two states of minimal elastic energy in a slender body. In this case, the energy barrier comes from the existence of a geometrical constraint: the system cannot transit from one state to the other without passing through an intermediate shape that generates in-plane stresses. We have seen that for thin bodies these in-plane stresses are associated with an important elastic energy cost. Elastic energy is then accumulated until it is so large that the barrier is crossed and the shape suddenly snaps through.

This principle is used by the carnivorous plant Dionaea muscipula (best known as the Venus flytrap) to trap insects^53,123,136  (Figure 1.12a). The trap of the Venus flytrap consists of two doubly curved, shell-like lobes, which are convex (curved outward) in the initial state. Because bending such a shell requires crossing a stretching energy barrier (see below), the trap can store elastic energy without closing. However, when the inner hairs are triggered, the plant actively changes the natural curvature of its lobes in one direction. This is enough to cross the elastic energy barrier, leading to a sudden change of curvature and closure.

It is possible to quantify this mechanism using a minimal one-dimensional elastic energy model and scaling arguments.⁵³  Let consider a shell of typical size L, thickness h and Young's modulus E. From Hooke's law, the total elastic energy of the shell scales as: , where V ∼ hL² is the volume of the shell and ɛ is the characteristic elastic strain associated to the deformation of the shell. When the shell is bent, two different modes of elastic deformation occur.⁸⁷  The first one is associated to pure bending, i.e. the fact that changing the curvature of a plate with respect to its natural curvature induces a gradient of elastic deformation within the thickness of the shell. The elastic strain associated to this bending mode is: ɛ_bend ∼ h(C − C⁰), where C is the actual curvature and C⁰ is the natural curvature. The corresponding bending elastic energy is thus:

The second mode of deformation is related to geometrical incompatibility: it is not possible to bend and reverse the curvature of a shell without stretching or compressing its surface. This is a direct consequence of the Gauss’ Theorema egregium,⁵  which implies that a surface cannot change its Gauss curvature without changing the natural length, or metric, of its surface. To estimate the stretching deformation ɛ_stretch induced by such a change of Gauss curvature, let consider a shallow shell of radius of curvature r ≫ L, which is deformed into a flat disc (Fig. 1.12b). Assuming that the length L along the diameter of the shell is conserved, the shell's perimeter must increase from πL₀, where L₀ is the initial projected diameter, to πL, so that ɛ_stretch ∼ (L − L₀)/L₀. Pythagoras's theorem implies that Δ² + (L$02$/4) = L²/4 and (r − Δ)² + (L$02$/4) = r². Since Δ ≪ r and Δ ≪ L, we have at first order: ɛ_stretch ∼ 2Δ²/L² ∼ L²/(32r²)∝L²κ$G0$, where κ$G0$ = 1/r² corresponds to the natural Gauss curvature of the shell. Here the Gauss curvature of the deformed state is zero since the disc is flat. This relationship can be generalized to a deformed state of arbitrary non-zero Gauss curvature as: ε_stretch ∼ L²(κ − κ$G0$), where κ_G = C² is the Gauss curvature of the deformed state. Therefore, the elastic energy associated to the stretching mode is given by:

The total elastic energy of the shell is the sum of the bending energy (1.77) and stretching energy (1.78): _el = _bend + _stretch. It is convenient to make the curvatures dimensionless using the characteristic curvature , such that: , . Using eqn (1.77)–(1.78), the dimensionless total elastic energy of the shell is then:

The parameter α quantifies the ratio of the stretching elastic energy to the bending elastic energy and depends only on the geometry of the shell: the thinner, wider or curved the shell, the greater the stretching energy compared to the bending energy. We see below that α determines how the shell flips from convex to concave when the natural curvature C⁰ is gradually changed. To this end, we note that, for a given value of C̃⁰ and α, the equilibrium shape of the shell C̃_eq is obtained by minimizing the total elastic energy with respect to C̃:

Figure 1.12c,d show qualitatively the shape of the total elastic energy as function of C̃ when the natural curvature of the shell C̃⁰ goes continuously from convex (C̃⁰ > 0) to concave (C̃⁰ < 0). For α ≪ 1, the bending energy dominates: the elastic energy exhibits only one minimum and the curvature of the shell at equilibrium is always very close to the natural curvature C_eq ≈ C⁰. Therefore, when the natural curvature changes from positive to negative, the shell shape follows the natural curvature and smoothly changes from convex to concave (Figure 1.12c). The situation is markedly different for α ≳ 1 (Figure 1.12d). In this case, the stretching energy is large and the total elastic energy (1.79) may exhibit two local minima separated by an energy barrier, depending on the value of the natural curvature. As shown in Fig. 1.12d, when the sign of the natural curvature changes, the system initially remains stuck in the first local minimum of convex shape, until the energy barrier disappears and the shell suddenly flips to the second minimum of energy, corresponding to the concave shape. Therefore, a small change of natural curvature (the internal motor of the motion) may lead to a very large change of actual curvature from convex to concave. This is the amplification mechanism used by the Venus flytrap and other carnivorous plants to speed up their motion and engulf their preys.^53,162 

The active process by which plants like the Venus flytrap trigger the instability and overcome the energy barrier is still not fully elucidated, raising interesting questions about signaling and fast mechanical actuation at the molecular level.^50,66,154  However, the general principle of using an elastic instability to overcome the hydraulic limit and amplify the speed of motion is robust and weakly depends on these microscopic details.¹⁴⁶  This strategy has already inspired several artificial devices, such as fast soft actuators and jumping robots.^73,79,94,168  More generally, the non-muscular movements of plants offer a wealth of mechanisms worth studying from a physics and engineering perspective.^19,22,140  Biomimetic applications of plant movements are further discussed in Chapter 8.

I would like to thank the whole scientific community of Plant Biomech’ and GDR PhyP (“Plant Biophysics and biomechanics”, CNRS no 2007), from whom I learned most of the ideas and concepts discussed in this chapter. I particularly thank Jacques Dumais and Bruno Moulia who introduced me to the fascinating field of plant biomechanics almost 20 years ago. This work was supported by the European Research Council (ERC)under the European Union's Horizon 2020 research and innovation program (Grant 647384).