The ratio of the saturation water vapor concentration (w_i) (or saturation water vapor pressure; for an ideal gas these are interchangeable) over a liquid-gas interface to that over an interface with water potential of zero (w_i,0) is (Nobel, 2005)

where ψ is the water potential, V_H20 is the molar volume of water (18 × 10⁻⁶ m³ mol⁻¹), R is the universal gas constant and T the interfacial temperature (for various symbols see Table 1). In case of water at water potential of zero, i.e., pure water at atmospheric pressure, the ratio is one and the saturation vapor concentration (w_i,0) depends only on temperature (Nobel, 2005).

Transpiration rate (E), is proportional to the difference between the water vapor concentration in the sub-stomatal cavity w_i and water vapor concentration in the ambient air w_a (2)E=(wi-wa)g where g is stomatal conductance (Nobel, 2005). Inserting Equation (1) into Equation (2) and introducing the saturation ratio of the ambient air (S≡wawi,0; RH being 100% × S) the transpiration rate is expressed as

For the flat surface exp(ψV_H2O/RT) = 1 and for negative water potentials the term is less than one (see Figure 2) and that facilitates the situation that the transpiration (E) turns to negative when S > exp(ψV_H2O/RT). The actual transpiration rate in relation to the transpiration rate at a leaf water potential of zero (E₀) is then

Also boundary layer conductance contributes to leaf water exchange, but here it is included in the stomatal conductance term g.

Saturation vapor concentration of CO₂ over a surface of water at water potential of zero is commonly expressed using Henry's law (5)caq,0=ciHCO2 where c_i and c_aq,0 are the leaf CO₂ concentrations in the air and aqueous phases, and H_CO2 is the Henry's law coefficient for CO₂ in water, which is dependent on temperature and pH (Nobel, 2005). If the decrease in water potential at the site of evaporation is caused solely by loss of water (and not osmotic concentration) then the Henry's law's relation in Equation (5) is modified so that (Vehkamäki, 2006)

where γ is surface tension of water, r is the radius of curvature (defined positive for a concave surface), V_CO2 is the partial molar volume of CO₂ in water (34 × 10⁻⁶ m³ mol⁻¹), T_N is the water pressure difference over the air-water interface (T_N = P_air − P_liquid, where P_air is air pressure and P_liquid is liquid pressure), and where the Young-Laplace equation (T_N = 2γ/r) is used for the relation between interface curvature and surface tension. Note that in physics r is typically defined negative for a concave surface but we follow here the convention often used in eco-physiology of plants. Equation (6a) means that the higher the T_N, the lower the saturation vapor pressure, and hence the larger the partitioning to the aqueous phase (the larger “effective Henry's law constant,” H_CO2 times the exponential term) of CO₂ in water. Since the osmotic concentration of the apoplastic water is generally assumed to be small (Nobel, 2005), we assume in the following calculations that the water potential at air-water interface is lowered only by T_N and the osmotic component of water potential is negligible, i.e., −ψ = 2γ/r, leading to: (6b)caq=ciHCO2 exp(ψVCO2RT)

The driving force for stomatal gas exchange of CO₂ is the difference between the CO₂ concentration in ambient air and the CO₂ concentration at the site of photosynthesis in the chloroplasts c_c. The pathway of CO₂ movement is divided into air and aqueous phases. In steady state, CO₂ flux in the air phase  (JCO2air) (7)JCO2air=(ca-ci)gCO2air must equal the aqueous phase flux (JCO2aq) (8)JCO2aq=(caq-cc)gCO2aq where gCO2air and gCO2aq are the air and aqueous phase diffusive conductances, respectively. Both fluxes must also equal the net assimilation rate (A) in steady state, which we approximate to be linearly proportional to CO₂ concentration in the chloroplasts (e.g., Mäkelä et al., 1996) (9)A=fcc where f is a constant of proportionality depending on light availability, temperature, and biochemical properties of the photosynthetic machinery. The relation between CO₂ assimilation rate and chloroplasts is assumed linear in order that an analytical solution can be obtained for assimilation rate as a function of leaf water potential. Note that in reality, A saturates with increasing c_c (this becomes more evident at higher CO₂ concentrations), and also the compensation point of CO₂ is involved in the classical Farquhar formulation of the photosynthesis rate (Von Caemmerer and Farquhar, 1981). However, Equation (9) is a reasonable assumption over any small range of c_c and over the CO₂ limited region of the A–c_c curve (Lambers et al., 2008).

Combining equations (6b) to (9) reveals that the relative increase in CO₂ assimilation rate resulting from the reduced vapor pressure of CO₂ can be expressed simply in terms of the ratio between internal (air phase) and ambient CO₂ concentration (see Supplementary Material for the algebraic derivation)

This formulation shows that the values for the resistances to CO₂ diffusion, light level and the absolute value of the Henry's law coefficient need not to be considered to evaluate the relative effect of water potential on the CO₂ assimilation rate. Further calculations (see Supplementary Material) reveal that the time scale required for the steady state CO₂ concentration to be reached in the leaves after changes in water potential is less than 1 s so the steady state approach used here is well-justified.

Note that Equation (10) does not contain the effective Henry's law coefficient H_CO2 including the dissociation of CO₂ into HCO3- at higher pH values (Taiz and Zeiger, 2002), and the aqueous phase diffusive conductance gCO2aq. They are quantities of which estimation requires information beyond the modeling framework here. The aqueous pathway of CO₂ from the mesophyll surface into the chloroplast stroma is more complex than portrayed here, affected by e.g., lipid phase diffusive steps or different macromolecules obstructing diffusion (Taiz and Zeiger, 2002; Nobel, 2005).

The final note is related to assuming osmotic effects negligible. For dilute solutions the equilibrium water vapor pressure is lowered according to the Raoult's law, i.e., the equilibrium pressure of the pure water must be multiplied by the mole fraction of water in the solution. This creates negative water potential facilitating the negative transpiration similarly to a non-flat surface. Then wi=wi,0XH2Oexp(-2γVH2OrRT) (compare with Equation 1) where X_H2O is the mole fraction of water in the solution and is less than one. For CO₂ the situation is more complicated and one cannot say generally whether the osmotic effect lowers or increases the equilibrium vapor pressure. This depends on the type of solutes and their interactions with the dissolved CO₂ and water molecules.

Figure 1 concludes schematically the consequences of the existence of the curved interface for transpiration and carbon assimilation.

Finally, we define the water-use efficiency (WUE) as the ratio of the net assimilation and the transpiration rate as (12)WUE=AE

Ignoring the reduction of vapor pressure with decreasing water potential will also cause an error in the estimation of the value of stomatal conductance and leaf internal CO₂ concentrations from measurements of leaf gas exchange and vapor pressure deficit (VPD), especially when the measurements are conducted at high RH and low xylem water potential. Using Equation 3 for the actual stomatal conductance (g) and introducing the apparent stomatal conductance (g_app), i.e., the stomatal conductance if the reduction in water vapor pressure is ignored

the relationship between two conductances can be written as

The error made in the estimation of stomatal conductance will also propagate into the calculation of the internal CO₂ concentration (c_i). Formulating the CO₂ exchange by means of the two conductances as

where c_i,app is the internal CO₂ concentration, the ratio of the difference of the ambient CO₂ concentration and the internal CO₂ concentration to the difference against the apparent internal CO₂ concentration can be written as

Decreasing leaf water potential decreases the saturation vapor concentration of both water and CO₂, albeit more so in the latter than the former (Figure 2). This is because the partial molar volume of CO₂ in the water solution is larger than that of the molar volume of water (see Table 1). Decreasing water potential lowers the CO₂ saturation vapor pressure resulting in enhanced partitioning of CO₂ to water. For example, at water potential of −0.5, −2.0, and −10.0 MPa, the partitioning of CO₂ to water is 100.6, 102.8, and 114.6%, respectively, compared to that at water potential of zero, based on Equation (6b). Typical values of leaf water potential for C₃ plants are between −1 and −2 MPa (Mencuccini, 2003), down to −4 MPa in species in arid zones, and as low as −10 MPa in the most extreme cases (Tyree, 1997).”

The transpiration rate decreases as the leaf water potential decreases and the ambient RH increases. In the case that water potential is very low and RH is very high transpiration is predicted to turn into water uptake (see Equation 4). The “reverse transpiration” occurs along the zero contour in Figure 3A, being approximately at 94% and −10 MPa, and 98.9% at −2 MPa.

The increase in CO₂ assimilation rate with decreasing water potential becomes more pronounced with increasing internal CO₂ concentration (Figure 3B), i.e., under conditions of high stomatal conductance and/or low light. The daytime ratio of internal to ambient CO₂ concentration varies typically from 0.5 to 1 (e.g., Steudle, 2001), and even at low internal concentrations CO₂ (half of the ambient one), CO₂ assimilation rate increases by 1.7 and 8.7% for water potentials of −2.0 and −10 MPa, respectively. At water potentials of −10 MPa many plants will close their stomata (e.g., Pockman and Sperry, 2000). Naumburg et al. (2004) have reported non-zero stomatal conductances down to −7 MPa and Garcia-Forner et al. (2016) down to −5 MPa.

WUE, the ratio of assimilation and transpiration rates, increases with decreasing water potential and increasing RH and internal CO₂ concentration (Figure 4) as a result of the simultaneous decrease in transpiration rate and increase in CO₂ assimilation rate. For example, let us consider representative conditions at temperate/boreal zone: Intermediate RH of 50%, internal CO₂ concentration ratio of 0.7 and leaf water potential of −2 MPa. At these conditions WUE increases approximately 5% due to the decreased leaf water potential. The increase in WUE would naturally be more pronounced at lower leaf water potentials and higher RH, being 28%, if the potential was −5 MPa and RH 80% with the CO₂ concentration ratio remaining 0.7. However, one should note that the stomatal closure ignored here would partly mask the reduced vapor pressure effect.

If the reduction in vapor pressure due to decreasing water potential is not taken into account, the calculation of stomatal conductance from transpiration and ambient air VPD will yield an underestimation of the actual stomatal conductance (Figure 5A). The error becomes noticeable when water potential is low and RH is high. The error in the estimation of stomatal conductance also propagates to the estimation of leaf internal CO₂ concentration (Figure 5B). The actual value of leaf internal CO₂ concentration is then underestimated.

We analyse the case of a coastal redwood tree (Sequoia sempervirens) since it grows under the conditions where water uptake by leaves is known to be significant and the relevant information is available from several references. We consider a fog event during a dry season when RH is 100% (Burgess and Dawson, 2004) and temperature is 20°C. This means that the saturation vapor concentration over a surface of water potential zero (w_{i, 0}) is 0.92 mol m⁻³ (Haynes, 2010). The realistic leaf water potential values can be assumed to vary from −0.5 to −2.2 MPa (Ambrose et al., 2016). It is difficult to assess what the actual value for stomatal conductance would be during such conditions due to the difficulty in measuring the actual stomatal conductance (see Figure 5A) and also due to technical limitations of measuring small transpiration rates at high RH, when the transpiration may be also masked by condensation on the surfaces (e.g., Altimir et al., 2006). The maximum stomatal conductance for coastal redwood is likely to be around 5 mm s⁻¹ (Ambrose et al., 2009). Caird et al. (2007) compiled values of nocturnal stomatal conductance measured across many different species and the largest values of c. 5 mm s⁻¹ were for deciduous trees and shrubs. Increased stomatal conductance due to foggy conditions has been reported in many studies (Dawson et al., 2007; Reinhardt and Smith, 2008; Alvarado-Barrientos et al., 2015). We assume that the realistic conductance values vary from 1 to 5 mm s⁻¹. By using Equation (3) with the fixed value of RH (100%) and the temperature (20°C) and the above-mentioned ranges for the water potential and stomatal conductance we obtain the range of the reverse transpiration rates shown in Figure 6. If the conductance is 2.5 mm s⁻¹ and the potential −1.5 MPa, the reverse transpiration is 0.5 × 10⁻³ gm⁻²s⁻¹. The reverse transpiration may be short-lived if it equilibrates the leaves. However, the obtained estimate for the reverse transpiration rate is of the same order of magnitude as the measured downwards sap flow in these trees (1.5 L h⁻¹, which equals 0.6 × 10⁻³ g m⁻² s⁻¹ given a total leaf area of 660 m² (Burgess and Dawson, 2004). This sap flow rate is equal to ~5–7% of maximum daytime transpiration values (Burgess and Dawson, 2004). Transpiration will remain negative as long as RH is higher than 99% given a leaf water potential of −1.5 MPa.

The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.