Mechanical basis for thermonastic movements of cold-hardy Rhododendron leaves

Abstract

The profusion of rhododendrons in cold climates is as remarkable as the beauty of their blooms. The cold-hardiness of some of the montane species is in part due to reversible leaf movements triggered under frigid conditions wherein the leaves droop at the leaf stalks (petioles) and their margins roll up around the midrib. We probe the mechanics of these movements using leaf dissection studies that reveal that the through-thickness differential expansion necessary for leaf rolling is anisotropically distributed transverse to and along the midrib. Numerical simulations and theoretical analyses of bilayer laminae show that the longitudinal expansion amplifies the transverse rolling extent. The curvature diversion scales with the in-plane Poisson’s ratio, suitably aided by the stiff midrib that serves as a symmetry breaking constraint that controls the competition between the longitudinal and transverse rolling. Comparison of leaf rolling with and without the petiole indicates that the petiole flexibility and leaf rolling are in part mechanically coupled responses, implicating the hydraulic pathways that maintain the critical level of midrib stiffness necessary to support the longitudinal expansion. The study highlights the importance of curvature diversion for efficient nastic and tropic leaf movements that enhance cold-hardiness and drought resistance, and for morphing more general hinged laminae.

1. Introduction

Curling and rolling of Rhododendron leaves are indicators of the leaf temperature—the colder the temperature the greater their extent—and it is for this reason that rhododendrons are commonly referred to as nature’s thermometers. This unique set of reversible leaf movements confers a significant advantage to the plant for survival during the cold winter as the leaves are able to function despite the threats of snow accumulation, diurnal freeze–thaw cycles, and desiccating winds [1–5]. These movements were first reported by the British naturalist J. D. Hooker as part of his Himalayan expeditions [1]. At the onset of winter frost, he observed that the leaves of some of the more cold-hardy rhododendrons became pendant and rolled inwards, likely ‘from the expansion of the frozen fluids in the layer of cells on the upper surface of leaf, which is exposed to the greatest cold of radiation’. These shrubs are thus able to avoid dormancy under conditions that are ‘far beyond the ordinary limits of phanerogamic vegetation’ [1], a defining feature that helped spawn the rhododendron craze within the British Isles in the late nineteenth century.

More generally, the biological and ecological significance of nastic (plant-oriented) and tropic (stimuli-oriented) plant movements continues to be an active area of interest [6,7]. Of late, there is considerable activity in identifying the underlying biomechanical principles. Mechanical instabilities often become important in triggering these movements. Some recent examples include touch sensitivity in Venus flytraps [8] and Mimosa leaves [9], edge and in-plane growth during blooming of flowers [10,11], rolling of doubly curved grass blades [12], fluttering of growing leaves [13], and mechanical instabilities that drive seed dispersal [14]. Characterization of the differential strains effected through cellular-scale growth and/or changes in turgor pressure as well as their relation to the movements serve as a gateway for understanding the biological design principles and the biosignalling pathways, aspects that are important for comparative and evolutionary studies [15,16] and for (genetic) engineering of environmental robustness such as drought resistance [17] and cold-hardiness [18,19].

While there have been significant advances in understanding the bioecological significance of the leaf movements in cold-hardy rhododendrons [20–23], their mechanical basis remains unclear. The hydration of the petiole is implicated in the curling of the leaves, and the rolling response is loosely attributed to differential expansion due to a combination of cellular-scale turgidity gradients and intracellular freezing, fed by local water redistribution and possibly correlated with the leaf venation [5,24,25]. Here, we use a combination of leaf dissection experiments, numerical simulations and theoretical analyses to uncover the mechanical forces at play and their biological implications.

2. Methods

2.1. Freeze–thaw experiments

Our experiments are performed on leaves of Rhododendron maximum shrubs, a cold-hardy non-deciduous species that grows along the eastern Appalachian mountains. One-year-old leaves growing on outer canopy branches in the Jefferson National Forest, Blacksburg, Virginia (latitude 37.2852; longitude −80.4609) were excised at the base of the petiole with a razor blade, immediately enclosed in a sealed plastic bag, and transported to the laboratory. Leaves were kept fully hydrated, equilibrated in a freezer set for − 7°C, and then removed and immediately photographed (NIKON CoolPix 990 and Canon EOS 7D digital cameras).

2.2. Curvature extraction

The leaf curvatures were extracted using two separate techniques: (i) measuring the diameter 2r of the rolled leaves, approximated as part of a circular cross-section, at multiple points along the leaf and then analysing the profiles using a public domain image analysis program (Image J, http://rsbweb.nih.gov/ij/docs/all-notes.html) and (ii) by measuring the edge–edge distance between margins and the local leaf width, at various points along the leaf. In the former case, the curvature is simply κ_y = 1/r, and in the latter case, it is calculated by approximating the edge–edge distance and local leaf contour width as chord and arc lengths of a circle and then numerically solving for the curvature of the circle. The average over multiple curvature measurements is reported along with the standard error.

2.3. Leaf dissection experiments

2.3.1. Transverse and longitudinal strips

Strips from the leaf lamina with various thicknesses and orientations were excised depending on the rolling experiment being performed. Note that the isolated strips do not have access to the petiole-midrib hydraulic pathways that can enhance the abaxial–adaxial differential expansion. Then, the curvatures they develop on freezing are naturally lower bounds on the spontaneous curvatures expected for whole leaves. To minimize the water loss from the strip edges or because of their detachment from the midvein, the strips were hydrated before freezing. The lamina strips were then organized on white paper and placed in a freezer set for − 7°C. After equilibration, they were removed from the freezer and immediately photographed edge-on. The strip curvatures were extracted by analysing the images (jpeg format) using Image J.

2.3.2. Angled strips

Dissection studies were also performed for strips cut at three different angles to the midrib (electronic supplementary material, figure S3), α = 60°, α = 45° and α = 20°. On freezing, these strips both bend and twist indicating that the curvature tensor and therefore the spontaneous curvature tensor have a non-zero twist component along the direction of the cut direction, κ_xy ≠ 0 and ${\kappa }_{xy}^0\ne 0$. The twist is larger for angles close to α = 45° and is smaller for the other two angles. This twist component is absent for the transverse and longitudinal strips. Together, the trends indicate that these two directions are the principal directions associated with the spontaneous curvature tensor.

2.3.3. Leaf thickness measurements

The average leaf thickness is based on analysis of three separate leaf cross-sections. One centimetre wide transverse strips are excised from each leaf and preserved in FAA (50% ethanol, 35% water, 10% formalin, 5% acetic acid). Following complete preservation, sections were hydrated to remove the FAA, dehydrated in an alcohol gradient, saturated with xylene, impregnated with paraffin oil and equilibrated in hot paraffin. Ten-micrometre thick sections of the strips broken into five 1 cm sections were taken using an HM 340E rotary microtome (Microm International). Slides were double stained with safranin and fast green before permanent mounting. Leaf thickness was measured every 2 mm distance from midvein to the margin.

2.4. Numerical simulations

2.4.1. Geometry

The model leaf is a composite consisting of two semi-elliptical elastic shells that serve as the laminae, bonded to a cylindrical elastic midrib (figure 3a). Based on experimental characterization of healthy one-year-old R. maximum leaves, we take the semi-axes lengths of the leaves to be 2a = 21 cm and 2b = 5.5 cm. In order to simplify the analyses, spatial variations in the thickness of the laminae and the midrib diameter are ignored, and the average values of laminae thickness t = 0.36 mm and midrib diameter d = 1.05 mm are employed based on thickness measurements on transverse strips (see above).

Figure 3.

Numerical simulations of model leaves. (a) Top and side views of the model leaf. (inset) The abaxial–adaxial distribution of the expansion coefficient α that leads to differential expansion (see Methods). (b–d) Simulated equilibrium morphologies of the model leaf with varying spontaneous curvatures ${\kappa }_x^0$ and ${\kappa }_y^0$, and relative midrib moduli $\hat{E}=E_m/E$. Colour-scale from blue to red indicates increasing out-of-plane deflection. (e) Cross-sectional profiles of the rolled morphologies for varying spontaneous curvatures and midrib moduli. Black, blue and red curves correspond to the configurations shown in (b–d). The profile for ${\kappa }_x^0=0$ is shown for comparison (orange curve). (f) Rolling extent extracted as the equilibrium transverse curvature κ_y plotted as a function of longitudinal spontaneous curvature ${\kappa }_x^0$ for three different $\hat{E}$. The dotted vertical lines are critical points ${\kappa }_{xc}^0$ beyond which the leaf rolls longitudinally.

2.4.2. Differential expansion

The rolling differential expansion is induced by structuring the laminae as two layers of thickness t/2 bonded at the mid-surface with a no slip constraint. The relevant mechanical properties of the two layers (E, ν) are identical. The adaxial ice drives the differential expansion on freezing of the leaf surfaces, and to model its effect the adaxial layer is prescribed a negative expansion coefficient, − α. The freezing also drives the apoplastic water movements away from the abaxial aside. To model the ensuing changes in hydration on the abaxial side, it is prescribed an equal and opposite expansion coefficient + α. Then, for a given change in the overall leaf temperature ΔT, the local spontaneous curvature simplifies to κ⁰ = 3αΔT/t [26]. This relation is used to parametrize the anisotropic distribution of the expansion coefficient (α_x, α_y) with the experimentally extracted spontaneous curvature distribution (${\kappa }_x^0$, ${\kappa }_y^0$). For the ranges of spontaneous curvatures in this study, α_x and α_y are of the order of 10⁻⁴ (°C)⁻¹.

2.5. Finite-element simulations

The equilibrium shape of the discretized composite is extracted using a finite-element solver (ABAQUS, v. 6.10). The leaf is modelled using four-node doubly curved general-purpose shell elements with finite membrane strains (S4 in ABAQUS notation). The midrib is modelled using two-node linear beam elements (B31 in ABAQUS notation). A uniform mesh with an element size 0.2 cm was used in all of the calculations. The differential expansion is implemented within the user subroutine UEXPAN.

3. Results

3.1. Freeze–thaw experiments

Figure 1a shows the pendant and rolled leaves on a frozen rosebay rhododendron shrub (R. maximum). Freshly excised leaves under ambient conditions are elliptical and on average flat with a straight midrib (figure 1b). On freezing at − 7°C, the laminae roll around the straight midrib to shield the abaxial surface and the margins overlap completely. The leaf apex is slightly inflected, evident when viewed along the leaf tip (inset, figure 1c).

Figure 1.

Freezing-induced leaf movements in Rhododendron maximum. (a) Pendant and rolled leaves at sub-zero air temperatures. (b,c) Adaxial and abaxial surfaces of a freshly excised leaf at (b) room and (c) sub-zero temperatures (− 7°C). (c, right). The rolling (transverse) curvature ${\overline{\kappa }}_y$ averaged at several locations along the midrib for seven different leaves. The accompanying graphic is a distal–proximal view of one of the rolled leaf margins that shows a slightly inflected tip. (d, left) Abaxial view of the unrolling of a frozen leaf on exposure to room temperature. (d, right) Evolution of the edge–edge distance at midsection (solid red line) and the transverse curvature (solid black line). Scale bar, 1 cm.

Figure 1c shows a plot of the rolling extent, or the transverse curvature ${\overline{\kappa }}_y$ for seven frozen leaves of R. maximum averaged over multiple points along the midrib. The leaf curvature is comparable across the samples, with an average rolling extent of ${\overline{\kappa }}_y=1.58\pm 0.08\hspace{0.17em}{\text{cm}}^{-1}$. The thermonastic or sensitivity response is unlike that in drying leaves (electronic supplementary material, figure S1) wherein the edges scroll inwards, and it is completely reversible [27]. The frozen leaf unrolls completely in minutes on exposure to room temperature (figure 1d). The edge-to-edge distance at the midsection increases rapidly initially and then slowly saturates as the leaf flattens out. The rolling extent, also plotted in figure 1d, decreases accordingly. The trend is consistent with the strong correlation between leaf rolling and its surface temperature [3,21].

3.2. Leaf-dissection experiments

We first characterize the mechanical state of the rolled leaf using dissection experiments (see Methods). Figure 2a shows the edge-on shape of a fully hydrated 2 mm wide transverse strip cut along the midsection before and after freezing to − 7°C. The strip bends non-uniformly with a larger curvature near the midrib and the twist is negligible. The spontaneous curvature for strips dissected at various points along the midrib is plotted in figure 2b. Averaging over multiple leaves yields a spontaneous curvature ${\kappa }_y^0=0.86\pm 0.22\hspace{0.17em}{\text{cm}}^{-1}$ that is significantly smaller than the rolling extent κ_y (figure 1c), indicating that the through-thickness differential expansion is insufficient to stabilize the rolled shapes.

Figure 2.

Spontaneous curvature distribution in rolled leaves. (a) Curvature development in a 2 mm wide transverse strip of a healthy leaf on freezing to − 7°C. (b, right) Variation of the curvature of transverse strips cut along the midrib ${\kappa }_y^0(x)$ as indicated (left). (c) Same as in (a), but for longitudinal strips of widths in the range 5–10 mm dissected parallel to the midrib for one-half of a healthy leaf (labelled S1–S4). The rolled strips are viewed edge-on and the adaxial side of each strip is indicated. The morphologies of the midrib (labelled M) and unsectioned half of the leaf (labelled S0) after freezing are also shown for comparison. Scale bar, 1 cm. (d) Mediolateral distribution of the longitudinal curvature ${\kappa }_x^0(y)$ after freezing to − 7°C in two leaves (solid and dashed lines) with one-half of the leaf dissected into five strips labelled S1–S5. The curvature of the midribs M is much smaller and not plotted.

Abaxial–adaxial water redistribution should result in a component of the differential strain along the midrib ε_xx and therefore a longitudinal spontaneous curvature ${\kappa }_x^0$. For isotropic expansion, we expect ${\kappa }_x^0={\kappa }_y^0$. The combination of the cellular-scale structure and venation architecture can lead to anisotropy in the freezing pathways that drives the differential expansion, and can result in deviations from the isotropic limit [21,23]. Leaf dissection parallel to the midrib reveals that while the bare midrib remains straight on average (labelled M, figure 2b), the longitudinal strips develop a curvature that increases towards the apex (labelled S1–S4). As is the case for transverse strips, the twist is negligible, ${\kappa }_{xy}^0\approx 0$. On the other hand, strips dissected at varying angles to the midrib bend and twist on freezing (electronic supplementary material, figure S2), confirming that the transverse and longitudinal directions are the principal directions associated with the spontaneous curvature tensor, $[{\kappa }^0]\equiv ({\kappa }_x^0,{\kappa }_y^0)$. The unsectioned other half of the lamina (labelled S0) rolls as expected, although its extent is measurably reduced possibly due to its detachment from the midvein. The strip curvature, averaged over several leaves, decreases towards the margins. The mediolateral gradient ${\kappa }_x^0(y)$ is plotted in figure 2d for two leaf samples with one-half of each leaf now sectioned into five strips (S1–S5) for better resolution. Averaging over the distribution for 10 leaves from two different shrubs, plotted in electronic supplementary material, figure S3, yields ${\overline{\kappa }}_x^0=0.59\pm 0.12\hspace{0.17em}{\text{cm}}^{-1}$ that is measurably smaller than its transverse counterpart.

The anisotropy in the expansion is clearly significant. While both spontaneous curvatures exhibit gradients along the transverse and longitudinal directions, our dissection studies show that the mediolateral gradient ${\kappa }_x^0(y)$ represents the dominant contribution to the anisotropy. We, therefore, average out the other gradients as constant, global (spontaneous) curvatures, and investigate the effect of the anisotropy using a combination of numerical simulations and theoretical models.

3.3. Numerical simulations

We first employ finite-element-based numerical simulations of model leaves to study its effect on the rolling extent (Methods). The leafy laminae are simplified as semi-elliptical elastic plates (axes dimensions 2a = 21 cm, 2b = 5.5 cm) of constant thickness t = 0.36 mm, connected to a cylindrical elastic midrib of average diameter d = 1.05 mm. The dimensions correspond to year old healthy leaves of R. maximum. The leathery scleromorphic laminae in these evergreens have Young’s modulus in the range E ∼ 50−100 MPa while the midrib is stiffer, E_m ∼ 600−1200 MPa [28,29], and to make contact we explore a wider range of (relative) midrib modulii $\hat{E}=E_m/E=0-10$. The form of differential expansion extracted in these leaves is implemented by prescribing an in-plane distribution of the abaxial–adaxial expansion coefficient [α] ≡ (α_x, α_y) that also sets the spontaneous curvature $[{\kappa }^0]\equiv ({\kappa }_x^0,{\kappa }_y^0)$ (inset, figure 3a).

The interplay between the differential expansion, midrib modulus and equilibrium morphology of the model leaf is shown in figure 3b–d. The longitudinal curvature distribution ${\kappa }_x^0=f(y)$ is a nonlinear fit to the experimentally extracted mediolateral variation (figure 2; electronic supplementary material, figure S3). At small transverse curvatures ${\kappa }_y^0\le 0.5\hspace{0.17em}{\text{cm}}^{-1}$ and in the absence of a midrib ($\hat{E}=0$), the leaf rolls longitudinally (figure 3b). Increasing the relative midrib stiffness to $\hat{E}=1$ or increasing the transverse curvature to ${\kappa }_y^0=0.6\hspace{0.17em}{\text{cm}}^{-1}$ results in a transition to transverse rolling (figure 3c,d, respectively). Figure 3e shows the cross-sectional profiles through the midsections of the simulated morphologies. For fixed relative midrib modulus and transverse expansion ($\hat{E}=1$, ${\kappa }_y^0=0.6\hspace{0.17em}{\text{cm}}^{-1}$) the edge–edge distance decreases from 90% for ${\kappa }_x^0=0$ to $45\text{\%}$ for ${\kappa }_x^0=f(y)$, and the rolling extent increases from κ_y = 0.6 cm⁻¹ to κ_y = 0.75 cm⁻¹. Simulations with systematic increase in the average longitudinal expansion ${\kappa }_x^0$ confirm that this is a general trend (figure 3f). In particular, the rolling extent κ_y increases linearly until a critical point ${\kappa }_{xc}^0$ beyond which we observe a transition to longitudinal rolling. Simulations with varying midrib moduli indicate that a stiffer midrib delays the transition by increasing ${\kappa }_{xc}^0$, thereby elevating the capacity of the leaf to absorb longitudinal expansion. Interestingly, it does not directly contribute to the rolling extent ${\kappa }_y({\kappa }_x^0)$, also observed in the transverse profiles shown in figure 3e. Evidently, the anisotropy in the expansion is crucial for enhancing the rolling extent, aided by the stiff midrib that selects the rolling direction.

3.4. Theoretical analysis

To quantify this interplay, we turn to a theoretical framework based on elastic plates composed of elliptical thin shells bonded to a cylindrical midrib, as in the simulations, and subject to abaxial–adaxial expansion (electronic supplementary material, Methods). The bending and stretching of the composite plate is described by Föppl–von Kármán equations that relate the in-plane (membranal) strains and forces to out-of-plane deflection w(x, y) and related curvatures [κ] [30]. We approximate the model leaf as an orthotropic linear elastic shell with uniform curvatures [31–34]. It is stiff along the long (x)-axis with a modulus ratio E_x/E_y that is energetically equivalent to that of a composite leaf with relative modulus E_m/E. The extensional and bending rigidities [S] and [D] scale with the corresponding rigidities of isotropic thin shells, S = Et/(1 − ν²) and D = Et³/12(1 − ν²) with ν the in-plane Poisson’s ratio, while the midrib rigidities are that of a cylindrical beam, S^m = E^md²/4b and D^m = E^md⁴/64b (figure 3a). In-plane and out-of-plane equilibrium, together with the strain compatibility that couples changes in the Gaussian curvature Δκ_G = κ_xκ_y to membranal distortions, and vanishing forces and moments all enforced along the edge of the bilayer yield the contributions of the laminae and the midrib to stretching and bending energy densities, $u_s([ϵ])=([S]\hspace{0.17em}{[ϵ]}^2+S^m{ϵ}_x^2)/2$ and $u_b([\kappa ])=([D]{[\kappa -{\kappa }^0]}^2+D^m{\kappa }_x^2)/2$ (electronic supplementary material). The equilibrium shape of the shell is a stationary point in the elastic energy functional of the composite $U={\int }_A(u_s+u_b)\hspace{0.17em}\mathrm{d}A$ integrated over the elliptical planform.

The evolution of the elastic energy with increasing expansion (${\kappa }_x^0,{\kappa }_y^0$) is depicted in figure 4a,c, for relative midrib modulus $\hat{E}=1$ and fixed spontaneous curvature ratio ${\kappa }_x^0/{\kappa }_y^0=0.8$. The trends are qualitatively the same for other values of this ratio. At small levels of expansion, the energy is minimized by a doubly curved shape with both κ_x and κ_y slightly less than their respective spontaneous values (figure 4a). The positive Gaussian curvature κ_G > 0 implies that the shape change results in in-plane stretching of these thin laminae [30,32,35]. As the expansion increases, the transverse curvature κ_y increases at the expense of its longitudinal counterpart κ_x and eventually exceeds the spontaneous transverse curvature, ${\kappa }_y>{\kappa }_y^0$ (figure 4b). Beyond a critical expansion, transverse rolling dominates with vanishing longitudinal and (therefore) Gaussian curvatures κ_G ≈ 0, indicating that the laminae recover their initial stretch and the rolling thereafter is isometric (figure 4c). We also see appearance of secondary minima corresponding to longitudinal rolling (${\kappa }_x>{\kappa }_x^0>0,{\kappa }_y=0$) separated from the primary minima by an energetic barrier that itself increases with expansion. The numerically simulated equilibrium shapes shown below each energy plot are qualitatively similar.

Figure 4.

Theoretical analyses of leaf rolling dynamics and stability. (a–c) The energy $\hat{U}=U/U_m$ of an orthotropic elliptical shell as a function of curvature (κ_x, κ_y), for fixed midrib modulus $\hat{E}=1$ and spontaneous curvature ratio ${\kappa }_y^0/{\kappa }_x^0=0.8$. The prescribed spontaneous curvatures increase from (a) to (c). Red circles denote the minima for transverse rolling (equilibria at U = U_m) while the black circle in (c) is a local minima associated with longitudinal rolling. Simulated equilibrium shapes are shown below each panel. The colour-scale corresponds to the out-of-plane deflection. (d) Stability diagram associated with the shape evolution in (a–c). The black and red lines show the equilibrium longitudinal and transverse curvatures, respectively, as a function of the longitudinal expansion ${\kappa }_y^0$. The dashed lines correspond to longitudinal rolling. The white circles denote equilibrium curvatures in numerical simulations as indicated. (e) Phase diagram showing the effect of midrib stiffness on the rolling direction for an inextensible plate with varying degrees of anisotropic expansion, ${\kappa }_x^0/{\kappa }_y^0$. The light and dark shaded regions correspond to transverse and longitudinal rolling, respectively. The critical spontaneous curvatures ${\kappa }_{xc}^0(\hat{D})$ extracted from numerical simulations at fixed ${\kappa }_y^0=0.5\hspace{0.17em}{\text{cm}}^{-1}$ are plotted for comparison. The green shaded region corresponds to scleromorophic evergreen leaves with dimensions of healthy R. maximum leaves.

The theoretical predictions for the equilibrium curvature tensor are summarized in figure 4d as a shape stability diagram, [κ]([κ⁰]). For a direct comparison, the equilibrium curvatures of the simulated morphologies in figure 4a,c are also indicated. The competition between the two rolling directions nascent in the initial doubly curved shape tilts in favour of a transversely rolled shape and, as in the numerical simulations (figure 3e), we see a linear increase in the transverse curvature. The curvature evolution for the secondary minima associated with longitudinal rolling is qualitatively similar in that κ_x increases linearly. Past the initial transient, κ_G ≈ 0 and the isometric rolling extent can be estimated as the inextensible limit of the uniform curvature framework by minimization of the transverse bending energy $U_b^t=U_b({\kappa }_x=0)$ (electronic supplementary material, Note):

$${\kappa }_y={\kappa }_y^0+\nu {\kappa }_x^0.$$ 3.1

The prediction is consistent with parametric simulations of equilibrium shapes with varying in-plane Poisson’s ratio (electronic supplementary material, figure S4).¹ Using the average values for spontaneous curvatures in frozen R. maximum leaves, ${\overline{\kappa }}_y^0\approx 0.9\hspace{0.17em}{\text{cm}}^{-1}$ and ${\overline{\kappa }}_x^0\approx 0.6\hspace{0.17em}{\text{cm}}^{-1}$, and a Poisson’s ratio of ν = 0.5, we predict a rolling extent of κ_y ≈ 1.2 cm⁻¹ that approaches the extracted rolling extent (figure 1c). This phenomenon of curvature diversion has also been observed in past studies on thin ribbon-like laminae subject to isotropic spontaneous curvatures, differential strains and edge stresses [14,36–40].

The inextensible analysis also sheds light on the role of the midrib in controlling the rolling direction and the distribution of differential expansion. The critical longitudinal expansion ${\kappa }_{xc}^0$ is the point at which bending energies associated with transverse and longitudinal rolling become equal $U_b^t=U_b^l$:

$$\frac{{\overline{\kappa }}_{xc}^0}{{\overline{\kappa }}_y^0}=\frac{\nu \hat{D}+(1-{\nu }^2){(1+\hat{D})}^{1/2}}{1-{\nu }^2(1+\hat{D})},$$ 3.2

where $\hat{D}=D^m/D$ is the relative midrib stiffness (electronic supplementary material, Note). The prediction together with the simulation results is plotted in figure 4e for relative midrib stiffnesses in the range $0<\hat{D}<1$. For healthy R. maximum leaves, the midrib is moderately stiff relative to the laminae with $\hat{D}\approx 0.8$, where we have used a relative midrib modulus of $\hat{E}\approx 5$. Then, ${\kappa }_{xc}^0=1.75\hspace{0.17em}{\kappa }_y^0$ and using ${\kappa }_y^0=0.9\hspace{0.17em}{\text{cm}}^{-1}$ based on the dissection studies, the critical point ${\kappa }_{xc}^0\approx 1.6\hspace{0.17em}{\text{cm}}^{-1}$ is well above the longitudinal expansion extracted from the dissection experiments. The inflected apex of the rolled leaf suggests a local transition to longitudinal rolling, a combined effect of the larger longitudinal expansion near the midrib in the leaf dissection studies (figure 2e) and a decrease in the midrib diameter and therefore its stiffness. Evidently, the distribution of adaxial–abaxial expansion in the remainder of the leaf is tuned to enhance transverse rolling.

4. Discussion and conclusion

The leaves of some of the cold-hardy rhododendrons are characterized by denser and thicker epidermis and palisade layers [41]. Based on our results, it is likely that the underlying cellular scale adaptations are tailored to increasing Poisson’s ratio necessary for the curvature diversion, in addition to facilitating the water redistribution that simultaneously drives the differential expansion and maintains a critical level of midrib stiffness. Stomata closure under frigid conditions precludes long-range water transport [42], implicating local redistribution from petiole. The hypothesis is consistent with hydration changes at the petioles that become pendant prior to the rolling and straighten up after the laminae fully unroll, and also with our experiments on rolling extents in leaves excised with and without petioles. For R. maximum, the rolling extent decreases from ${\overline{\kappa }}_y=1.69\pm 0.55\hspace{0.17em}{\text{cm}}^{-1}$ for leaves with petioles to ${\overline{\kappa }}_y=1.33\pm 0.49$ without petioles (electronic supplementary material, figure S5). Similar experiments on leaves of R. catawbiense, another cold-hardy species, result in a decrease from ${\overline{\kappa }}_y=2.07\pm 0.41\hspace{0.17em}{\text{cm}}^{-1}$ with petioles to ${\overline{\kappa }}_y=1.75\pm 0.40$ without petioles, i.e. the rolling of excised leaves without petioles is almost always suppressed. Then, although the bioecological significances of these two leaf movements are distinct [21], their mechanics is likely coupled via water transport from the petiole that feeds into freezing pathways from the midrib to the ground tissues, that in turn drives the differential expansion.

The mechanical insight offers clues into the biological basis for these leave movements, in particular the cellular-scale architectural adaptations and biosignalling pathways that mediate the freezing pathways that drive the differential expansion. In particular, the phenomenon of curvature diversion we uncover here has implications for the movements of naturally occurring hinged laminae such as leaves and their modified cousins, flowers, as well as for the function of insect wings and related aerodynamic architectures. The integrated understanding serves as an enabler for engineering cold-hardiness in genetically modified crops, and for biomimetic strategies aimed at shaping their synthetic counterparts within tessellated origami structures and microelectronic devices.

Data accessibility

All data generated or analysed are available in the paper or its electronic supplementary material.

Authors' contributions

E.N. and M.U. performed the leaf dissection experiments; H.W. performed the numerical simulations; H.W. and M.U. developed the theory. All authors analysed the results, contributed to writing the manuscript. M.U. conceived of the study, designed the study, coordinated the study and helped draft the manuscript. All authors gave final approval for publication and agree to be held accountable for the work performed therein.

Competing interests

We declare we have no competing interests.

Funding

H.W. acknowledges support from the Thousand Youth Talents Plan of China and the Fundamental Research Funds for the Central Universities (grant no. WK2090050042). M.U. was partially funded by the NSFDMREF programme (award no. 1434824).