A novel method of measuring leaf epidermis and mesophyll stiffness shows the ubiquitous nature of the sandwich structure of leaf laminas in broad-leaved angiosperm species

Highlight Leaf laminas are designed in sandwich structure, a very efficient structure used in e.g. airplane wing, which enables leaves to be thin, flat, ideal for photosynthesis but also mechanically stable.


Introduction
The primary function of plant leaves is recognized as photosynthesis and has been studied intensively from various points of view (Lambers et al., 2008;Blankenship, 2014). However, it is much less recognised that a large fraction (i.e. 14-77%) of leaf dry mass is in structural components i.e. cell walls (Onoda et al., 2011). The large investment in structural mass is considered to physically protect the photosynthetic leaf function from a suite of stressors including gravity, wind and herbivory (Read and Stokes 2006). Elucidating how the leaf anatomical structure is built in relation to these basic physical requirements is essential to better understand the functioning of leaves.
Leaves typically have a flat, thin structure, which is associated with a large leaf surface area per unit biomass, and is thus ideal for efficient light interception (Givnish, 1986;Braybrook and Kuhlemeier, 2010), but are concomitantly prone to mechanical failure due to their thin structure (Niklas, 1999;Read and Stokes, 2006). Therefore leaves should be designed in a way that they are not only thin but also reasonably stiff and strong. In engineering, such demands are typically met by designing objects as sandwich structures.
A typical sandwich structure is composed of stiff outer surfaces and a lightweight core, which greatly increases specific stiffness (stiffness per unit mass) in bending and has been used in many engineering constructions such as airplane wings and surfboards (Gibson et al., 1988;Gere and Timoschenko, 1999). Leaf laminas resemble a sandwich structure in that they are composed of two layers of epidermis tissue and intervening mesophyll tissues (Vincent, 1982;Gibson et al., 1988;Niklas, 1991Niklas, , 1992Niklas, , 1999Moulia et al., 1994;Moulia and Fournier, 1997). This design may be an evolutionary solution to increase lamina robustness with minimum biomass investment. It has been shown that the lamina (as well as veins) plays a significant role in whole leaf mechanical stability. For instance, Moulia et al. (1994) reported in maize that the lamina itself contributed about 50% to leaf bending stiffness at the middle of leaves and more towards the leaf tips. Similar importance of lamina stiffness in leaf bending stiffness was shown by a finite element method (Kobayashi et al., 2000(Kobayashi et al., , 2002. In addition to the importance of leaf laminas in maintaining whole leaf plane structure, a sandwich structure has also advantages in protecting leaves from herbivory and pathogen attack by its stiff and strong surface (Grubb, 1986).
Even though the sandwich structure of leaf laminas has been alluded to in several studies and its potential advantage for plants is recognized (Vincent, 1982;Gibson et al., 1988;Niklas 1991Niklas , 1992Niklas , 1999Moulia et al., 1994;Moulia and Fournier, 1997;Niinemets and Fleck, 2002), to our knowledge no study has actually determined to what extent leaf laminas are indeed designed as a sandwich structure. It is therefore also unknown how common this design is among different plant species. Such an assessment requires knowledge of the mechanical properties of the epidermis and mesophyll layers. However, we know of only one study that evaluated the mechanical properties of the epidermis and mesophyll layers (Gibson et al., 1988). They indirectly estimated the Young's modulus (intrinsic material stiffness) of each layer by assuming stiffness of the epidermis and mesophyll layers to be proportional to their respective cell wall cross-sectional areas in Iris leaves. The lack of knowledge for stiffness of leaf epidermis and mesophyll layers may be due to the practical difficulty in isolating leaf epidermis layers from other tissue. Furthermore, separating tissues from a turgid leaf may release tissue stress (also called residual stress in engineering), and could have confounding effects on the estimate of Young's modulus of tissues (Hejnowicz and Sievers, 1995a).
In this study, we first quantified the extent to which leaf laminas behave as sandwich structures. To do so, we developed a novel method to quantify the Young's moduli of the epidermis and mesophyll layers without separating these layers, based on linear elastic theory (Gere and Timoschenko, 1999). As discussed later, biological materials are much more complicated than an ideal material that follows linear elastic theory (Baskin and Jensen, 2013), yet we believe that this approach is an important first step to evaluate the mechanical properties of the epidermis and mesophyll layers. By applying this method to a diverse set of 36 species, we determined to what extent sandwich structures are common across species (i.e. herbaceous, woody deciduous or woody evergreen) and across phylogeny. Furthermore, the variation in the Young's moduli of the epidermis and mesophyll layers was analysed with respect to leaf morphology (e.g thickness and tissue density) and anatomy (e.g. mesophyll, epidermis, cuticle and cell walls) to better understand how leaf structures are functionally built in relation to the multiple requirements for photosynthesis, mechanical stability and defence.

Model
In the present study, we use a few material terms that may not be commonly used in plant science, so first we explain these terms to avoid possible confusions. Young's modulus is an intrinsic mechanical property of the material independent of its geometry (see equation 9 for the calculation). Tensile stiffness and bending stiffness are structural properties of a material, which depend on the Young's modulus and geometry of the material (see also equations 1 and 2, Gere and Timoschenko 1999).
Many leaves have a flat plane structure consisting of two layers of epidermis and an intervening mesophyll core, which could be approximated by a sandwich structure as described below. Some other leaves have more complicated structures such as cylinder or shell structure, which require a more complicated mechanical modelling and are not considered in this study.
For a simple sandwich structure, such as a plane leaf lamina, longitudinal tensile stiffness of the sandwich structure in the tensile test can be expressed as a sum of the longitudinal Young's moduli of the face (E f ) and core (E c ) weighted by cross-section area of each layer, according to the superposition principle under the linear elastic theory (Gere and Timoschenko, 1999) (Fig. 1).
where A is cross-section area of the composite sandwich structure, and A f and A c are respectively the cross-section area of the face (i.e. epidermis) and core (i.e. mesophyll) layers (A=A f +A c ). Similarly, bending stiffness of the sandwich structure can be expressed as a sum of E f and E c weighted by the second moment of area of each layer (Gere and Timoschenko, 1999). The second moment of area is a geometrical property of a beam and depends on cross-section area and shape (described later).
where I is the second moment area of the composite sandwich structure, and I f and I c are the second moment of area of the epidermis and mesophyll layers relative to the neutral axis of the leaf (I=I f +I c ). A c /A is equal to relative thickness of the mesophyll in the whole lamina thickness, α (0<α<1, Fig. 1) and A f /A is thus equal to 1-α. From these, equation 1 can be rewritten as follows: For the bending test, the second moment of area of a layer (full and symmetric relative to the neutral axis) is proportional to the third power of its thickness (Gere and Timoschenko, 1999). Therefore the second moment of area of the mesophyll layer (I c ) is equal to α 3 I, Equations 7 and 8 show that Young's moduli of the epidermis layers (E f ) and that of the mesophyll layer (E c ) can be estimated from only three measurable variables: (i) Young's modulus of the lamina measured by tensile tests (E T ), (ii) Young's modulus of the lamina measured by bending tests (E B ) and (iii) the fraction of lamina crosssection composed of the mesophyll layer (α), which can be determined microscopically (see Methods section 'Anatomy analysis'). One may argue that when the bulk core material is too compliant, there is a possibility of buckling and subsequent delamination of the faces upon bending. This effect may be negligible in leaves since the mesophyll tissues are turgid and the bundle sheath extensions, which resemble a honeycomb structure, keep a distance between the two epidermis layers.

Plant materials
We collected leaves from 36 broad-leaved species (33 different families) including 15 herbaceous species, 12 deciduous woody species and 9 evergreen woody species (Table 1) that were grown outdoors in the Utrecht University Botanical Garden (latitude=52.087N, longitude=5.167W) on 13 September 2006. All sampled leaves were fully developed without any visual damage or senescence. These species were collected to cover a wide range of angiosperm species including magnoliids, monocots and a wide variety of eudicots ( Supplementary Fig. S1). However very small, highly dissected or twisted leaves were avoided as these leaves could not be used for the mechanical measurements. Each sampled leaf was wrapped in a wet paper towel, and five or more leaves per species were combined and sealed in a plastic bag, and stored at 4°C to avoid loss of turgor pressure until just before the mechanical measurements.

Mechanical tests
From each sampled leaf, three intercostal lamina strips in total were excised avoiding primary veins (and also secondary veins when they were very distinct) by a pair of razor blades that were fixed in parallel and 0.52 cm apart. These samples were then used for the following mechanical tests; one strip was used for the tensile test, and the other two for the bending test on both the adaxial and abaxial sides of the leaves. Each lamina strip was excised just before the measurement so that the test specimen was kept as fresh as possible. When leaves were too small to excise enough strips, adjacent leaves were also used. There were five replicates per species for each measurement. One set of replicates (tensile and bending tests of 36 species) was completed in one day. In total five consecutive days were required to complete the measurements of five replications of 36 species. Neither apparent deterioration nor systematic change in leaf mechanical properties was observed during the measurement period.
The tensile and bending tests were conducted with a general testing machine (5542, INSTRON, Canton, MA, USA) at room temperature (~23°C) and relative humidity (50-70%). In the tensile tests, the specimens were clamped by a pair of pneumatically controlled grips, and the free length between the clamps was ~5 cm. Before the measurements, width (w) and free length (l 0 ) of the strips were measured by a caliper and the thickness at the middle of the strips (d) was measured by a thickness gage (7313, Mitsutoyo, Japan). Tension was applied at a constant speed of 25 mm min -1 . Force (F, N) and displacement (δ, mm) were measured every 100 milliseconds until the specimen was torn ( Supplementary Fig. S2). The Young's modulus in the tensile tests (E T ) was estimated from the initial slope of the relationship between force (F) and displacement (δ) based on the following relationship (Gere and Timoschenko, 1999) (Fig. 1): where A is the cross-section area of the specimen and l o is the original length of the specimen between the clamps. Tensile strength (the maximum force per unit cross-section area of the specimen) was also recorded.
In the bending tests, a leaf strip was placed on two supports 15 mm apart, and force was applied at the middle of the specimen at a speed of 25 mm min -1 . Force (F) and displacement (δ) were measured as in the tensile test ( Supplementary Fig. S2). The Young's modulus in the bending test (E B ) was estimated from the initial slope of the relationship between force and displacement based on the following relationship (Gere and Timoschenko, 1999) (Fig. 1): where L is the span length and I is the second moment of area of the specimen. In a specimen with a rectangular cross-section, I is typically calculated as 1/12wd 3 , where w is width and d is thickness of specimen. The span-thickness ratio was reasonably high (53 ± 20), thus shear deformation was negligible. Adaxial E B and abaxial E B measures were strongly correlated with each other (R 2 =0.97) with a slope close to 1 (slope=1.024, 95% confidence interval of the slope=0.95-1.10, standardised major axis slope; Warton et al., 2006). Therefore adaxial E B and abaxial E B were combined and averaged per each leaf sample, and used for the calculations of E f and E c .

Anatomy analysis
Small segments (∼1 × 2 mm) of lamina were excised for anatomical analyses from all measured leaves (n=5 for each species). The segments were fixed with 2.5% glutaraldehyde in 100 mM phosphate buffer (pH=7.0) and kept at 4°C until analysis. Leaf segments were sliced at a thickness of ∼10 μm and stained with Nile blue, a lipophilic dye. UV fluorescence images of the section were taken with a microscope (AX-LH 100 Olympus Optical, Japan) and used to measure thicknesses of each layer (epidermis, palisade, spongy parenchyma, cuticle, and cell walls of the epidermis) with image analysis software (analySIS, Olympus, Japan). Intercellular airspace was calculated from leaf fresh mass, dry mass and thickness with an assumption that specific gravity of water and leaf solid tissue were 1 and 1.5 g cm -3 respectively (Roderick et al., 1999).

Calculation and statistical analysis
The Young's moduli of the epidermis layers (E f ) and mesophyll layer (E c ) were calculated with equations 7 and 8. In this study, 'epidermis layer' means epidermis tissues plus cuticle membranes, and 'mesophyll layer' means intervening tissues between the epidermis layers (mostly mesophyll cells and some minor veins). Data were log-transformed before the statistical analysis when the data were not normally distributed (Shapiro-Wilk test, P<0.05). Differences in the values among functional groups were tested with ANOVA. Multiple comparisons were done with a Bonferroni correction. Pearson's test was used to test correlation (r) among traits. Standardized major axis slope (Warton et al., 2006) Filled circles denote the optimal α that maximizes E B /E T at a given β. Note that when the epidermis layer is extremely thin and stiff compared to the mesophyll layer (α≈1, β≈0), E B /E T approaches to the value of 3.
(see Supplementary Fig. S1). The branch length was assumed to be a constant since detailed divergence time for each species was not clear. Moran's autocorrelation index was used to test whether there was a phylogenetic autocorrelation in traits (Moran, 1950;Paradis, 2012). Phylogenetically independent contrasts (PICs; Felsenstein, 1985) were used to test whether correlations among traits in the cross-species comparisons were driven by coordinated evolutionary trait-shifts in a convergent manner across the phylogeny. All statistical analyses were performed with the R software package (v3.0.1, R Foundation for Statistical Computing, Vienna, Austria).

Results
The Young's moduli measured by tensile tests (E T ) or bending tests (E B ) were positively correlated to lamina tissue density, meaning that stiff leaves were made of dense tissues (Fig. 3A). The Young's moduli measured by bending tests (E B ) were consistently higher than E T across 36 species, meaning that not only tissue density but also tissue arrangement, i.e. sandwich structure, affect the Young's modulus when it is measured in bending (Fig. 3A). There was a strong correlation between E T and E B across species (R 2 =0.93, P<0.0001, Fig. 3B Abbreviations are as follows; FG, functional group; H, herbaceous species; W, woody species; D, deciduous; E, evergreen; E T , Young's modulus measured by the tensile test (MPa); E B , Young's modulus measured by the bending test (MPa); Th, leaf thickness (mm); α, mesophyll thickness per unit lamina thickness (m m -1 ); E f , Young's modulus of epidermis layer (MPa); E c , Young's modulus of mesophyll layer (MPa); E B I, bending stiffness per unit lamina width (N mm 3 ). Means of five replications are shown.
The relationship between E B /E T and the mesophyll fraction (α) is shown in Fig. 5A. All species fell in the region of high E B /E T ratio (>1) and high α (close 1) as expected for sandwich structures. However, there were some species with higher E B /E T ratios than the theoretical maximum lines (1+α+α 2 , see equation 6), which can be achieved when β (=E c /E f ) equals 0. This deviation might be because (i) α was underestimated, (ii) there were some mechanisms beyond the linear elastic theory or (iii) other technical issues (see Discussion). When the inner part of epidermis layer was included in the mesophyll fraction (α'), most species fell within the theoretical lines (Fig. 5B).
The epidermis and mesophyll Young's moduli (E f and E c ) were calculated by applying the measured values of E B , E T and the mesophyll fraction (α) to equations 7 and 8. E f was much higher than E c , ranging from 11 MPa in Arabidopsis thaliana to 463 MPa in Eucalyptus pauciflora (Table 1; Fig. 6). On the other hand, E c was much lower (0.61 ± 9.61 MPa, n=36). The relatively large SD for the mean (i.e. high coefficient of variation) in E c was partly due to the structure of the data. For example, E c was calculated from E B and E T (equation 8) and therefore variation in E c was comparable in magnitude to the variation of E B and E T , whereas the mean value of E c was much smaller than those of E B and E T . While the accurate estimate of E c might not be possible in this method (see Discussion), qualitatively it is evident that the longitudinal stiffness of leaf laminas was largely determined by the stiffness of the epidermis layers. Among functional groups, E f was higher in evergreen woody species than in deciduous woody and herbaceous species (Fig. 6B, P<0.01).
The variation in the epidermis Young's modulus (E f ) was significantly correlated with the fraction of outer cell walls and cuticles within the epidermis layers (R 2 =0.56, P<0.001, n=36) (Fig. 7A). Each component, i.e. cuticle fraction or outer cell wall fraction in the epidermis layers, significantly correlated to the epidermis Young's modulus (P<0.001), but the cuticle fraction rather than the cell wall fraction correlated more strongly with the epidermis Young's modulus (R 2 =0.43-0.50 versus 0.13-0.23, Table 2). The thickness of the epidermis layer itself was not significantly correlated to the epidermis Young's modulus (Fig. 7B, Table 2, P>0.05), while leaf mass per area (LMA) was significantly correlated to the epidermis Young's modulus (Fig. 7C, Table 2, R 2 = 0.52). The Young's moduli of mesophyll layers were not significantly correlated with any leaf morphological or anatomical traits unless phylogeny was considered (Table 2). When phylogeny was considered, the Young's moduli of mesophyll layers were slightly negatively correlated to water content and positively correlated to the mesophyll fraction. Phylogenetic autocorrelations were weak and were not significant in most of the measured traits (Supplementary Table  S2). Furthermore, a consideration of phylogenetic divergences did not strongly alter levels of correlations among traits ( Table 2). The interpretation of these results is that the trait associations observed in this study were a result of repeated evolution across different clades of plants rather than a phylogenetic bias in the selection of studied species.

Discussion
In this study, we developed and applied a novel method to estimate the Young's moduli of epidermis and mesophyll layers. Our model based on linear elastic theory has a number of simplifying assumptions, and there were some phenomena that cannot be fully explained by this model (as discussed later). Yet, our approach showed that leaf laminas were designed as efficient sandwich structures in which the epidermis tissues were much stiffer and thinner than the mesophyll tissues. In addition, we showed that this sandwich structure was found in a wide range of studied angiosperm species, suggesting that a sandwich structure has general advantages for many land plants. In this discussion, we discuss anatomical and ecological considerations of leaf sandwich structures and also discuss future challenges including technical issues.

Anatomical considerations
Across species, the epidermis Young's modulus was strongly correlated with the fraction of cuticles and outer cell walls in the epidermis layer (Fig. 7), suggesting that the cuticles and epidermis outer cell walls played major roles in determining the surface stiffness. In particular, evergreen leaves had stiffer epidermises (Fig. 6) and thicker cuticles than deciduous leaves (Fig. 4), suggesting cuticles are mechanically important for long-lived leaves. This implication is also supported by studies on isolated leaf cuticle membranes (Wiedemann and Neinhuis, 1998;Bargel et al., 2006;Onoda et al., 2012), which found that leaf cuticles were made of very stiff material (50-1,500 MPa). Furthermore E f increased more than proportionally with the fraction of cuticles and outer cell walls in the epidermis layer, indicating that thicker cuticles were made of stiffer materials (Onoda et al., 2012).
The mesophyll tissues were much softer than the epidermis tissues, which should be due to the thin mesophyll cell walls, little longitudinal continuity and the presence of intercellular airspaces. Mesophyll cells generally have much thinner cell walls (0.05-0.4 μm, Terashima et al., 2011) than epidermis cells (1.4-9.2 μm, Fig. 4B). Mesophyll layers also contain larger intercellular airspaces; our calculation showed that 7-41% of the leaf volume was occupied by air (Supplementary Table S1). The thin cell walls of mesophyll and the presence of intercellular airspaces are primarily important to facilitate a high rate of CO 2 diffusion for photosynthesis (Parkhurst, 1977;Terashima et al., 2001), but our study suggests that air spaces can also contribute to higher bending stiffness (i.e., the product E B I) by increasing the second moment of area (I) in the sandwich structure.
There were minor veins in our test specimens, but their contribution to bending stiffness were likely small, as indicated by low E c . This was in part due to their location close to the neutral axis, i.e. under bending, minor veins located close to the neutral axis deform much less than outer tissues (i.e. epidermis). Thus minor veins geometrically cannot contribute much to E B . Such negligible contribution of minor veins to maintain a leaf horizontal position was also reported by a finite element analysis (Kobayashi et al., 2000(Kobayashi et al., , 2002. Studies on hypocotyls of Helianthus annuus (Asteraceae) also showed that 86% of tensile stiffness came from the epidermis (Hejnowicz and Sievers, 1995b). Similarly, in turgid tulip stems the epidermis was found to contribute up to 50% of stem bending stiffness (Niklas and Paolillo, 1997). Minor leaf veins are however important for water and photosynthate transport (Brodribb et al., 2007;Sack et al., 2013) and shear resistance (Lucas et al., 1991).

Ecological significance
Some leaves, especially herbaceous ones, are turgid i.e. hydrostatically inflated. They rely on turgor pressure to increase bending stiffness without much investment of resources in cell walls. It was intriguing that herbaceous species had significantly higher E B /E T values than woody species (2.80 ± 0.48 versus 2.45 ± 0.44, t-test, P<0.05) as well as higher water content (0.79 ± 0.10 versus 0.64 ± 0.07, t-test, P<0.05), suggesting that turgor pressure can increase efficiency of the sandwich structure. In turgid leaves, epidermis and mesophyll layers are maintained in states of tension and compression respectively because the hydrostatically inflated 'core' mesophyll is accommodated by the much stiffer 'face' epidermis layers (Hejnowicz and Sievers, 1995a;Kutschera and Niklas 2007). This 'pre-stressed sandwich structure' may increase efficiency of sandwich structure in at least two ways. First, the hydrostatic pressure increases lamina thickness and thus widens the distance between two epidermis layers. This increases the second moment of area of the epidermis layers in relation to the neutral axis and contributes to more efficient sandwich structure. Second, the epidermis tissue may be 'strain-stiffening material' (Hejnowicz and Sievers, 1996). While this phenomenon cannot be explained by the linear elastic theory, if the epidermis layers become stiffer under hydrostatic pressure, they can increase efficiency of a sandwich structure by decreasing β (=E c /E f ). Such mechanical efficiency should enable herbaceous plants to maintain a larger leaf area for a given biomass, which could contribute to them having relatively fast growth rates compared to woody species (Poorter et al., 2009), although other factors play a role as well. On the other hand, due to their reliance on water, hydrostatically inflated leaves tend to wilt sooner than leaves of woody species when water is limited. Leaf wilting is also an important function for plants as they can avoid strong sunlight and reduce water loss from leaves. Thus, the hydrostatically controlled-sandwich structure may be a quite efficient mechanism enabling plants to both acquire and avoid light energy depending on water availability. In terms of leaf mechanical stability, a sandwich structure may be particularly beneficial for plants growing under light limited conditions because they are often carbon-limited and produce thinner leaves (as compared to the same species growing under stronger light) to achieve high efficiency of light interception per unit biomass (Terashima et al., 2001;Poorter et al., 2009). These shaded leaves tend to have a larger fraction of intercellular airspace (e.g. Lee et al., 2000). Indeed in Plantago major, the fraction of intercellular airspace was found to increase by 28% with a shading treatment (Onoda et al., 2008). The relatively large airspace in shade leaves was more likely associated with demands on mechanical stability than with the necessity for rapid intracellular CO 2 diffusion for photosynthesis, since photosynthesis under shade is hardly limited by the intercellular CO 2 diffusion resistance (Terashima et al., 2001). Bending stiffness scales to the third power of lamina thickness and, for example, a 10% larger lamina thickness due to an increase intercellular space can increase lamina bending stiffness by 33% without adding biomass. Therefore a larger intercellular space is likely important to maintain larger light interception area for a given mass which is advantageous in shaded environments. Overall we observed a large interspecific variation in leaf lamina stiffness indices i.e. 36-fold in E B and 790-fold in bending stiffness (= E B × I) (Table 1), while the interspecific variation in fresh leaf mass per unit leaf area was much smaller (7-fold) (Supplementary Table S1). The much larger variation in bending stiffness than in fresh leaf mass may suggest that some leaves (especially evergreen leaves) may be too stiff (over-engineered) relative to the need to resist gravity or moderate wind, although an integrative approach including leaf weight, material properties and leaf dimensions is needed to Table 2

. Cross-species correlations (Pearson's test) and phylogenetic independent contrast (PIC) correlations among leaf traits across 36 species
Abbreviations: E f and E c are Young's modulus of epidermis and mesophyll layers respectively. E T and E B are Young's modulus measured by the tensile test and bending test respectively. Log 10 -transformation was applied when data were not normally distributed (Trans = 1). Bold letters indicate the level of significance (P<0.05).  understand whole leaf elastic stability (e.g. Fournier et al., 2013;Tadrist et al., 2014). These leaves were nevertheless designed as sandwich structures as indicated by their high E B /E T ratios, suggesting that a sandwich structure has benefits beyond being efficient in resisting gravitational and wind forces. Relatively thick cuticles may act as stiff and tough barriers against attacks from herbivores and microorganisms e.g. pathogens and bacteria (Grubb, 1986), and leaf toughness is a key element for longer leaf lifespan (Coley, 1983;Kitajima and Poorter, 2010;Onoda et al., 2011). Actually, in the present study, the Young's moduli of the epidermis layers were strongly correlated with LMA (leaf mass per area), which is a good indicator of leaf lifespan (Wright et al., 2004). These observations suggest that sandwich structures may be beneficial not only for maintaining leaf plane structure against gravity or moderate wind, but also for long lifespan by protecting leaf functions from various external stresses. While bending stiffness is important for leaves, leaves should be also deformable to reduce drag during strong winds (Vogel, 1989). This requirement may sound contradictory with the requirement for stiffness, but it may be possible because leaf surfaces (i.e. cuticles) are highly extensible materials in spite of their stiffness (Onoda et al., 2012). The requirement for deformation is likely to increase with leaf size because drag force is proportional to surface area (Vogel, 1989). In other words, small leaves do not necessarily deform much under wind. This may accord with the general pattern that very stiff leaves (e.g. sclerophyll leaves) tend to be small (Grubb, 1986).

Technical issues and future challenges
In this study, we show that the E B /E T ratio is a useful indicator of the extent to which leaves are designed as sandwich structures. Our method seems to be applicable to plant leaves since the E B /E T ratio of filter papers, which were more or less homogeneous material, was close to 1 (0.996 ± 0.17) while most leaves had the ratios much higher than 1 (2.59 ± 0.48, n=36). However, there were some species that exceeded the theoretical threshold of E B /E T (=3), albeit by a relatively small margin. Similarly, some mesophyll tissues had slightly negative Young's moduli while the Young's modulus should be a positive value. These may be artifacts possibly resulting from assumptions made in this study. We have at least three possible explanations for these phenomena. (i) The effective thickness of the stiff epidermis layer may be smaller than the total thickness of the epidermis layer. This is because the cuticle and outer cell walls largely determine the epidermis Young's modulus (Fig. 7), whereas the inner part of the epidermis may behave more like the mesophyll tissues (Fig. 5B). If so, our mesophyll Young's modulus values were underestimations and this could have resulted in negative values. Actually, recalculating the Young's modulus of the mesophyll layer in this way, two-thirds of species that initially had negative moduli switched to positive values. However, this issue cannot explain why some leaves had E B /E T values larger than 3 (Fig. 5). (ii) There was a possible technical issue with the measurement. If leaf laminas were not completely flat in a direction perpendicular to the longitudinal direction (although we tried to exclude non-flat leaves), the effective second moment of area could be higher than the calculated second moment of area that assumed a rectangular cross-section. If so, this would result in an over-estimation of E B while E T would be unaffected. Similarly, if lamina thickness was not homogeneous within a leaf specimen, a slight underestimation of lamina thickness might result in an over-estimation of E and this effect would have been larger in the bending test than in the tensile test. These technical issues can result E B /E T values >3 and, subsequently, an apparent negative mesophyll Young's modulus. (iii) As mentioned earlier, biological materials are quite complicated and do not always behave according to the linear elastic theory. It is known that soft tissues like mesophyll behave like 'Fung materials', which have a larger resistance to compressive strain than to tensile strain (Fung, 1993;Vandiver and Goriely, 2008). Furthermore, the epidermis tissue may be 'strain-stiffening material' (Hejnowicz and Sievers, 1996). There are also other effects including viscoelasticity, interactions between stress and moisture content, geometric non-linearity produced, e.g. by the Poisson's effect (Moulia, 2013). These effects together may lead to E B /E T >3 and, consequently, result in apparent negative mesophyll Young's moduli. While our aim was to understand the basic principle underlying the leaf mechanical design across species, these complicating factors cannot be ignored and could be important for mechanical stability of leaves.
In this study, we focused on leaf laminas and did not consider mechanical properties of major veins or petioles. The mechanical contribution of major veins to whole leaf bending stiffness is important, especially near the leaf base, while lamina stiffness has a major role in leaf bending stiffness toward the leaf tips (Moulia et al., 1994;Kobayashi et al., 2000Kobayashi et al., , 2002. Integrating the geometry and mechanical properties of laminas, veins and petioles will be an important challenge to understand mechanical stability of whole leaves (Tadrist et al., 2014).

Conclusions
Our novel approach indicates that many leaf laminas are designed as ideal sandwich structures, with the exclusively thin and stiff epidermis layers enveloping a thicker and softer mesophyll core. We showed that this design principle is commonly found in broad-leaves across a wide range of angiosperm taxa. This structure enables land plants to produce thin flat leaves with a high light interception per unit invested mass and yet resistant to buckling under gravitational forces and moderate wind. The stiff surface should also be important as a barrier against external biotic and abiotic stresses such as herbivory, pathogen attacks and rainfall. Anatomical features such as stiff cuticles, thin mesophyll cell walls, large intercellular airspace and hydrostatic pressure within leaves seem to be elegantly coordinated to optimize multi-functions of leaves, such as photosynthesis, mechanical stability and defence. These results provide new insights into the functional significance of leaf structure and have implications for plant sciences and biomimetic studies.

Supplementary material
Supplementary material is available at JXB online.
Supplementary Text S1. A full derivation of equations 7 and 8.
Supplementary Table S1. Additional leaf traits of 36 species.
Supplementary Table S2. Phylogenetic autocorrelations for measured traits across 36 species. Supplementary Fig. S1. A phylogenetic tree of 36 species.