Design of a unidirectional water valve in Tillandsia

The bromeliad Tillandsia landbeckii thrives in the Atacama desert of Chile using the fog captured by specialized leaf trichomes to satisfy its water needs. However, it is still unclear how the trichome of T. landbeckii and other Tillandsia species is able to absorb fine water droplets during intermittent fog events while also preventing evaporation when the plant is exposed to the desert’s hyperarid conditions. Here, we explain how a 5800-fold asymmetry in water conductance arises from a clever juxtaposition of a thick hygroscopic wall and a semipermeable membrane. While absorption is achieved by osmosis of liquid water, evaporation under dry external conditions shifts the liquid-gas interface forcing water to diffuse through the thick trichome wall in the vapor phase. We confirm this mechanism by fabricating artificial composite membranes mimicking the trichome structure. The reliance on intrinsic material properties instead of moving parts makes the trichome a promising basis for the development of microfluidics valves.

Drying of the thin water film on the surface of T. aeranthos following exposure to fog (the same leaf shown at 5 min. intervals). As the surface water evaporates, the leaf recovers its lighter aspect characteristic of dry trichomes. The surface water plays the role of a water reservoir for evaporation. All our measurements were done after this first phase of evaporation to ensure we were measuring the intrinsic resistance of the leaf. 30;31 as well as from molecular dynamics simulations in cellulose 32 (green circles). In the range relevant for our study (M < 40%), the data sets share a common trend despite the variation of methods (and pore microstructures): in all cases the diffusion coefficient decreases as the moisture content is reduced (i.e. as |Ψ| increases), following an empirical law A exp(αM ) where the coefficients for the lower (resp. upper) fits are α ≈ 0.12 % −1 and A = 1.6 · 10 −12 m 2 s −1 (resp. 4 · 10 −10 m 2 s −1 ) (domain in light blue, both limits are shown as dash lines). (b) Effective water diffusion coefficient in cellulose D cell from molecular dynamics simulations 32 showing the variation with the water potential |Ψ| = −Ψ. The dash line is a fit D cell ∼ 6.0 · 10 −5 |Ψ| −0.8 . The inset presents the same data function of the relative humidity, showing in light blue the domain corresponding to the same span than in the previous fits of panel (a).

Supplementary Note
This Supplementary Note presents the quantitative analyses necessary to infer the conductances associated with the different elements of the Tillandsia trichome.
Water potential, flux and conductance. The water potential Ψ is defined relative to a reference potential Ψ 0 which customarily is taken as the potential of pure water at standard temperature and pressure. Relative to this reference potential, the water potential of plant cells is typically negative and has two principal contributions. First, there is an entropic contribution from dissolved substances that gives rise to the osmotic potential Ψ Π = RT ln(γn)/V w where R the gas contant, T the temperature, V w the molar volume of water, γ the dimensionless activity coefficient and n the water mole fraction 33 . The water potential of the cells also includes a hydrostatic potential Ψ P corresponding to the hydrostatic pressure that arises from the elastic deformation of the cell walls and potentially the Laplace pressure associated with the numerous air-water interfaces in the interstices of cell walls. All together, the potential inside the cells of plants is given by the relation 34 : and typically ranges from to −1 to −0.1 MPa. Note that gravitational contributions to the water potential have been neglected. On the other hand, the water potential in the environment outside the leaf Ψ out depends on the temperature and relative humidity (RH) of the air: where R = 8.314 · 10 −6 m 3 MPa K −1 mol −1 and V w = 18 · 10 −6 m 3 mol −1 . For T = 293 • K and a relative humidity of RH = 50%, the water potential of the air Ψ out is equal to −94 MPa. For RH = 20%, Ψ out is as negative as −218 MPa. In contrast, air saturated with fog has a water potential of Ψ out ≈ Ψ 0 = 0 MPa.
When the water potential inside the cells of a plant (Ψ in ) differs from the outside potential (Ψ out ), a net water movement is expected from the region of high water potential to the region of low water potential. Under the assumption that the system is close to equilibrium, stationary and onedimensional, the mass flux density of water Q (g m −2 s −1 ) is proportional to the difference in water potential, ∆Ψ = Ψ out − Ψ in (MPa), across the membrane restricting the flow: The proportionality factor L (g m −2 s −1 MPa −1 ) is the conductance of the membrane to water flow (and is the inverse of the resistance R). It is sometimes more convenient to use the difference in water concentration ∆c (mol m −3 ) as the thermodynamic "force" driving flow instead of the difference in water potential. This is the case when dealing with the vapor flux arising during transpiration where the one-dimensional first Fick's law yields: where D (m 2 s −1 ) is the water diffusion coefficient in the medium under consideration, δ is the distance over which diffusion takes place, and M w = 18 g mol −1 is the molar mass of water.
In case of a flux of vapor, equations (3) and (4) are equivalent, and one can identify the conductance of the membrane: where we used ∆Ψ ≈ RT Vw ∆c c following Ψ = RT Vw ln(RH). Note that c is the average concentration within the medium, that can be chosen as c = (c ext + c in )/2.
Liquid water absorption. The liquid water present on the surface of a Tillandsia leaf must cross successively the thick outer walls of the trichome and the plasma membrane of the foot cells before it can reach the living mesophyll. Our experiments, however, already indicate that the shield walls offer little resistance to the inner flow of liquid water. Specifically, we have shown that shaved leaves only increase their rate of water absorption by 12.5 % as compared to intact leaves (Fig. 4d).This increase is not statistically significant as determined by a Student t-test (P> 0.2). The same conclusion is reached from our experiments with CaCl 2 solutions ( Fig. 3e). Since it is known that Tillandsia species acquire minerals and other nutrients through their trichomes 35 , we expect the osmotic solution used in these experiments to permeate the entire outer shield and come in direct contact with the plasma membrane of the outermost foot cell. The resistance to the outward flow of liquid water observed in these treatments is therefore imputable to the plasma membrane and indicate that the resistance measured for the inward flow of fog water by the leaf is the same as the resistance to the outward flow under an osmotic grandient (Fig. 3e). We conclude that the rate of inward or outward flow of liquid water is set by the permeability of the plasma membrane of the foot cells. Given this conclusion, the plasma membrane resistance can be calculated as R mem = (n tri a mem )∆Ψ abs /Q abs , where Q abs = (240 ± 60) mg min −1 m −2 is the water absorption rate measured on intact leaves, ∆Ψ abs = 1.2 MPa is the water potential difference, n tri = 1.75 · 10 8 m −2 is the trichome area density and a mem = (1.5±0.1)·10 −10 m 2 is the area of the plasma membrane of the foot cell ( Supplementary Fig. 4). The prefactor n tri a mem = 0.026 is a geometrical correction accounting for the fact that the total area of the foot cells' membranes accounts for only a small fraction of the total leaf area. Based on this equation, the membrane resistance is R mem = (1.3 ± 0.3) · 10 −4 MPa min m 2 mg −1 which is in good agreement with the range of resistance reported for other plant membranes 36 .
Water evaporation. Unlike absorption of liquid water, the paths for evaporation are potentially numerous and complex ( Supplementary Fig. 8).
Since we are interested in the specific transport properties of the trichome, we must evaluate the impact of other evaporation paths that may be acting in parallel with the trichome path. Two main parallel paths exist. First, a substantial fraction of the leaf water could be lost by stomatal transpiration. Schmitt and coworkers working with T. recurvata estimated that the conductance of the stomata on a per leaf basis is approximately six times the conductance of the trichomes 37 . However, Tillandsia species are CAM (Crassulacean Acid Metabolism) plants and, as such, respiration and the associated stomatal transpiration are observed only at night 37 . We also observed some evidence of diurnal fluctuations in the rate of evaporation (Fig. 2a, inset) but have not attempted to repeat the experiments of Schmitt et al. since all of our experimental treatments were done during the day and showed highly significant effects on the rate of evaporation without having to distinguish between the stomatal and trichome transpiration (Fig. 4).
Nonetheless, the reader should keep in mind that the evaporative fluxes reported for the intact leaves are an upper bound for the actual evaporation of the trichomes. Considering that the plants were respiring for 12 hours daily, the actual evaporative loss at the level of the trichomes could be four times lower than the value we reported.
The second path for evaporation is the leaf cuticle which is also slightly permeable to water (Supplementary Fig. 8). However, due to the relatively low diffusion coefficient of water in wax D wax, w = 5 · 10 −14 m 2 s −1 38 , and using a cuticle thickness of δ cut = (1.8 ± 0.4) µm ( Supplementary Fig. 4), the resistance of the cuticle is expected to be about 30 times greater than the resistance of the trichomes, which confirms its minor contribution to the total evaporative losses.
A boundary layer resistance is present in all of our experimental treatments ( Supplementary Fig. 8), although we have reduced its contribution to a minimum by maintaining air circulation around the leaf with the help of a fan. Using an estimated boundary layer thickness of δ bdr = 1 mm, one gets R bdr = (8.1 ± 4.3) · 10 −3 MPa min m 2 mg −1 34 . Thus, the boundary layer is expected to contribute less than 0.005% to the overall resistance. We can test this conclusion by measuring the evaporation rate of the free surface water on the leaf since the latter is controlled only by the boundary layer. We found an evaporation rate for the free surface water 360 times higher than the basal rate of evaporation for the leaf (Fig. 4d), thus confirming that the boundary layer is not the main factor controlling the rate of evaporative losses.
Given the considerations stated above, the resistance of the trichome shield can be estimated from the measured overall evaporative flux as R wall = (n tri a wall )∆Ψ eva /Q eva = (3.3 ± 2.0) MPa min m 2 mg −1 , where Q eva = −(3.2 ± 2.0) mg min −1 m −2 , ∆Ψ eva = −94 MPa −(−1.2 MPa) = −92.8 MPa, and, as before, a geometrical factor n tri a wall = (1.75 · 10 8 m −2 ) · (6.5 · 10 −10 m 2 ) is used to infer the intrinsic resistance of the trichome. From eq. (5), we deduce an effective diffusion coefficient for water vapor inside the trichome cellulosic wall equal to D wall = (1.1 ± 0.8) · 10 −9 m 2 s −1 , where we used c = 0.72 mol m −3 and a wall thickness of δ wall = (21.3 ± 3.6) µm. The measured diffusion coefficient (at RH ≈ 50%) is in good agreement with values published elsewhere in the literature 39;32 . Note that in series with the cellulose shield, the cell membrane is expected to contribute to less than 0.002% to the overall resistance of the trichome if we refer to the value of the cell membrane extracted in the "Liquid water absorption" subsection, R mem = (1.3 ± 0.3) · 10 −4 MPa min m 2 mg −1 .
In the discussion above, we considered the properties of cellulose at a fixed relative humidity RH = 50%. While eq. (3) is still valid, the value of the conductance L depends on the relative humidity: the effective diffusion coefficient of water in cellulosic materials D wall varies strongly with the water potential, as shown by the extensive literature on wood drying 40;31;27;28;29;30;41 . However, due to the diversity of cellulose and wood microstructures, measurements do not fall on a unique universal law for D wall (Supplementary Fig. 9a) 31;27;28;29;30;41 . Despite this diversity, these results share a common trend: the decrease of the diffusion coefficient (D wall ) with decreasing moisture content (M ) can be fitted by an exponential although the prefactor can vary by a factor 250 (blue domain in Supplementary Fig. 9a). Results from molecular dynamics simulations 32 lead to a similar relationship between D wall and M , at least for moderate water content (M < 40%).
In order to quantify the effect of water content, we use the latter data set on which the diffusion coefficient was measured simultaneously with the moisture content and the water potential 32 . Supplementary Fig. 9b shows that these measurements are in good agreement with a law D wall ≈ 6.0 · 10 −5 |Ψ| −0.8 . From the comparison with measurements on wood, this law is a lower bound for the diffusion coefficient, and an upper bound is expected by multiplying the prefactor by 250, i.e. D wall ≈ 1.5 · 10 −2 |Ψ| −0.8 (blue domain in Supplementary Fig. 9b). Finally, from this empirical law for D wall (Ψ), we are able to estimate the expected conductance of cellulosic material from eq. (5), using the same geometrical considerations developed above: with δ wall is the thickness of the shield wall. Using |Ψ| ≈ − RT Vw ln RH and c ≈ RH psat 2RT (p sat ≈ 2.34 kPa), this equation yields the domain displayed in the inset of Fig. 3f.
Comparison with biomimetic composite membranes. We repeated our absorption and evaporation experiments with a simple biomimetic sys-tem inspired by the Tillandsia trichome. The system is made of an osmotic NaCl solution (Ψ in = −4.5 MPa) separated from the environment by a semipermeable membrane and a cellulosic layer. For absorption of water through the semipermeable membrane only, a mass flow of Q abs = (2.3 ± 0.4) · 10 3 mg min −1 m −2 was measured, and since ∆Ψ abs = 4.5 MPa, the resistance to liquid water absorption is R abs = ∆Ψ abs /Q abs = (2.0 ± 0.3) · 10 −3 m 2 MPa min mg −1 . In evaporation experiments, a mass flow of Q eva = −(1.1 ± 0.2) · 10 4 mg min −1 m −2 was recorded, and since ∆Ψ eva = −94 MPa −(−4.5 MPa) = −89.5 MPa, R eva = (8.1±1.5)·10 −3 m 2 MPa min mg −1 . From the two resistance values, we calculate an asymmetry ratio for the semipermeable membrane alone of R eva /R abs = 4.1. In other words, a system equipped with only a semipermeable membrane shows very little transport asymmetry under normal conditions.
In a second series of experiments, we worked with composite structures made of a semipermeable membrane and a layer of cellulose. The cellulose layer was made of N (up to 100) sheets of printer paper with density ρ s = 75 g/m 2 , each sheet with a thickness of 100 µm. For these experiments, R abs = (1.8 ± 0.3) · 10 −3 m 2 MPa min mg −1 and R eva = (9.5 ± 1.8) · 10 −1 m 2 MPa min mg −1 , leading to an asymmetry ratio of R eva /R abs = 530, which is more than 100 times the asymmetry of the naked semipermeable membrane. Although this asymmetry remains modest compared to the Tillandsia trichome, it is likely that the asymmetry could be improved by using a more conductive semipermeable membrane and optimizing the layer of cellulose to maximize its resistance to water vapor diffusion.