Research
Effects of stomata clustering on leaf gas exchange Peter Lehmann* and Dani Or* Soil and Terrestrial Environmental Physics, ETH Zurich, Universit€atstrasse 16, 8092 Zurich, Switzerland
Summary Author for correspondence: Peter Lehmann Tel: +41 44 632 63 45 Email: [email protected] Received: 7 October 2014 Accepted: 17 March 2015
New Phytologist (2015) 207: 1015–1025 doi: 10.1111/nph.13442
Key words: gas diffusion, leaf conductance, spatial organization, stomata clustering, stomatal aperture.
A general theoretical framework for quantifying the stomatal clustering effects on leaf gaseous diffusive conductance was developed and tested. The theory accounts for stomatal spacing and interactions among ‘gaseous concentration shells’. The theory was tested using the unique measurements of Dow et al. (2014) that have shown lower leaf diffusive conductance for a genotype of Arabidopsis thaliana with clustered stomata relative to uniformly distributed stomata of similar size and density. The model accounts for gaseous diffusion: through stomatal pores; via concentration shells forming at pore apertures that vary with stomata spacing and are thus altered by clustering; and across the adjacent air boundary layer. Analytical approximations were derived and validated using a numerical model for 3D diffusion equation. Stomata clustering increases the interactions among concentration shells resulting in larger diffusive resistance that may reduce fluxes by 5–15%. A similar reduction in conductance was found for clusters formed by networks of veins. The study resolves ambiguities found in the literature concerning stomata end-corrections and stomatal shape, and provides a new stomata density threshold for diffusive interactions of overlapping vapor shells. The predicted reduction in gaseous exchange due to clustering, suggests that guard cell function is impaired, limiting stomatal aperture opening.
Introduction Studies have shown that leaf stomatal size and density evolved over geological timescales; these structural changes are interpreted (using gas diffusion theories) as refinements in plant regulation of gas exchange (Raven, 2002; Franks & Beerling, 2009; Berry et al., 2010; de Boer et al., 2011; Assouline & Or, 2013; Buckley & Schymanski, 2013). Stomata patterns on leaves have received much less attention than the dynamic function of stomata (Croxdale, 2000). Various studies have revealed that an uniform stomata pattern is linked to the ‘one cell spacing rule’ that defines a minimum spacing between adjacent stomata, often of several stomata aperture diameters (Korn, 1993; Larkin et al., 1997; Geisler et al., 2000). This ‘rule’ ensures that there are sufficient intervening cells for the proper function of guard cells whilst reducing the diffusive interactions among neighboring stomata (Bange, 1953; Franks & Casson, 2014). However, as already observed by Fellerer (1892) for Begonia, various plant species produce stomata arranged in clusters (Gan et al., 2010; listed 39 families of land plants). Stomata clustering may result from errors in the complex signaling for stomata initiation and in cell orientation and division (Yang & Sack, 1995; Bergmann & Sack, 2007; Peterson et al., 2010; Akita et al., 2013). Environmental signals may also regulate stomata development and patterning (Wang et al., 2007), such as signals from older to younger leaves to reduce stomata density in water-stressed plants (Serna & Fenoll, 1997), or the formation of *These authors contributed equally to this work. Ó 2015 The Authors New Phytologist Ó 2015 New Phytologist Trust
stomata clusters in certain plants grown in dry environments (Hoover, 1986). The formation of stomata clusters was posited to reflect an adaption to dry environments aimed at reducing water loss from plant leaves (Gan et al., 2010; Franks & Casson, 2014). Recently, Dow et al. (2014) measured the effect of stomatal clustering on maximum leaf conductance of various genotypes of Arabidopsis thaliana expressing different stomatal patterns. Their results show that diffusive conductance from leaves with clustered stomata was 60% lower relative to those with uniformly distributed stomata for densities (sum of abaxial and adaxial leaf surface) in the range of 300–450 stomata per mm2. Dow et al. (2014) hypothesized that the observed reduction is probably due to impaired function of the tightly spaced stomata guard cells that may prevent complete opening of the stomata. However, an additional potential effect of clustering may involve increased diffusive resistance within stomata clusters as postulated by Gan et al. (2010) and Franks & Casson (2014), and recently shown theoretically and experimentally for other evaporating porous surfaces by Lehmann & Or (2013). To quantify the potential effects of stomatal clustering on gaseous diffusive resistance, we developed a physically based analytical model that accounts for the main components of gas exchange resistance in distributed and clustered stomata patterns on plant leaves. The expressions developed provide a theoretical basis for predicting gaseous flux reduction due to different stomatal clustering scenarios, and indirectly provide a proper context for the physiological impairment of guard cell hypothesis. In Supporting Information Notes S1 the model is applied to quantify gas conductance for a New Phytologist (2015) 207: 1015–1025 1015 www.newphytologist.com
New Phytologist
1016 Research
different type of clustering defined by the network of veins, indicating that the conduction is reduced by this type of clustering as well.
Materials and Methods Theory of diffusive gas transport through stomata The diffusive gas flux F (expressed as moles per unit area per unit time) through stomata-perforated leaves depends on the driving force (i.e. a concentration difference along the diffusive pathway), and on the conductance g [mol L2 T1] of the various obstructions between sub-stomatal cavity and the ambient atmosphere. It is convenient to use diffusive resistance R (i.e. the reciprocal of conductance g), to express the total resistance as a series of different resistances as presented in Fig. 1. In the following, we quantify the resistance and conductance of these elements and their dependence on stomata size, shape and spatial arrangement. Note that we often discuss the various terms in relation to ‘vapor transport’ and transpiration, but the same expressions apply to CO2 uptake as well. Computing leaf conductance for gas transport We first consider gaseous diffusive conductance of a stomatal pore gSP (marked by index SP for ‘stomatal pore’) of depth d [L] and cross-sectional area A [L2], defined by Brown & Escombe (1900) as: 1 A D p r2 k n ¼ gSP ¼ n ¼ RSP d v d
Eqn 1
with presistance RSP, pore radius r [L] (same area A with ffiffiffiffiffiffiffiffiffi r ¼ A=p), stomatal density n [L2] and the ratio k [mol L1 T1] between vapor diffusion coefficient D [L2 T1]
and molar volume of air v [L3 mol1] (with k ~ 103 mol s1 m1 for water). Brown & Escombe (1900) considered an additional diffusive resistance at the ends of a pore related to converging and diverging concentration shells forming at pore apertures. Stomata spacing plays a role in limiting the lateral extent of adjacent vapor shells. We distinguish between the standard gas exchange formulation, which defines this resistance as a property of the stoma itself (the ‘end-correction’), and formulations that consider the effects of stomata spacing on diffusive resistance across ‘concentration shells’ (e.g. Bange, 1953; Schl€ under, 1988). This raises a certain ambiguity present in the literature concerning the proper representation of diffusive resistance related to ‘end-correction’, the role of stomata shape, and stomata spacing. Specifically, we wish to clarify (1) the number of end-corrections to be considered (at one or both ends of the stomata pore), the effects of (2) stomata shape (circular or elliptic), and the influence of (3) spacing s between adjacent stomata. In Fig. S2 we show how assumptions concerning stomata shape may lead to incorrect conclusions regarding the end-correction. In the following we summarize the various expressions used in the literature to quantify the ‘concentration shell’ resistance (i.e. for one concentration shell; for concentration shells at both sides the expressions must be doubled). Note that because we will discuss the transport of vapor (transpiration) in detail we will use in the following the specific term ‘vapor shell’ instead of the more general term ‘gaseous concentration shell’. We differentiate between approaches that consider the limited spacing and interaction between vapor shells (marked by index ‘VS’ for interacting vapor shells) and end-correction formulations that do not consider the spacing of the vapor shells (with index ‘end’ for end-correction). End-corrections (no interaction) Rend ¼
RBL
Flux per stoma
δ
s
RSP Rend
RLeaf
2r d
Fig. 1 Resistances for vapor diffusion through stomata-perforated leaves with resistance through the stomatal pore RSP, vapor shells forming at the end of each stomatal aperture RVS that may interact with each other (related to end-correction Rend, see ‘Discussion’ in the text), and boundary layer resistance RBL. The resistances of the vapor shells and the pore resistance are combined and lumped into leaf resistance RLeaf. The total resistance of leaf and atmosphere in series is controlled by four dimensions: stomatal depth d, pore radius r, spacing between stomata s and thickness of boundary layer d. New Phytologist (2015) 207: 1015–1025 www.newphytologist.com
Eqn 2(a)
(Brown & Escombe, 1900)
Rend ¼
RVS
1 1 ¼ gend 4r k n
Loge ð4a=bÞ 1 ¼ gend 2p a k n
Eqn 2(b)
(Parlange & Waggoner, 1970) Interacting vapor shells
RVS ¼
1 ¼ gVS
1 1 1 4r p s k n
Eqn 2(c)
(Bange, 1953) Ó 2015 The Authors New Phytologist Ó 2015 New Phytologist Trust
New Phytologist RVS
1 ¼ ¼ gVS
Research 1017
1 1 1 2p r p s k n
Eqn 2(d)
(Schl€ under, 1988) with spacing s between stomata and radius of major and minor axis of ellipsoidal-shaped stomata, a and b, respectively. Eqn 2(a) for the resistance of a vapor shell forming above a circular pore was proposed by Stefan (1882) and subsequently used by Brown & Escombe (1900); for elliptical stomatal pores with b a, Eqn 2(b) from Parlange & Waggoner (1970) is more accurate (see discussion in Notes S2). The equations derived by Bange (1953) and Schl€ under (1988) consider the effect of limited spacing between pores on the diffusive resistance through the vapor shells. The derivations of Bange and Schl€ under differ with respect to the shape of the flux source; this is a disk in Eqn 2(c) from Bange (1953), and a hemisphere in Eqn 2(d) from Schl€ under (1988). Interestingly, Eqn 2(c) converges to Brown and Escombe’s end-correction in Eqn 2(a) for large stomata spacing s. Rend and diffusive resistance of interacting vapor shells RVS can be related (Eqn 3a) to define a critical density nVS (Eqn 3b) where the effect of vapor (or gas) shell interactions becomes negligible (< 10% of the end-correction resistance): pffiffiffi RVS 4r 4r n ¼1 ¼1 ps Rend p
Eqn 3(a)
p 2 RVS 1 0:9 ! nVS 40r 160 r 2 Rend
Eqn 3(b)
for stomata pffiffiffi spacing s defined on a square grid with size s ¼ 1= n. For small stomata densities n (and small pore radius r), the second term on the RHS of Eqn 3(a) becomes small; hence, the gas diffusion model of Bange (1953) that considers interacting vapor shells converge to standard end-correction formulation. Details of the differences between the two expressions are discussed in the Notes S3 and explained in specific examples shown in Fig. S3. With a few exceptions, the effects of stomatal spacing s on diffusive conductance through interacting vapor shells forming around a stomatal pore is rarely considered (Bange, 1953; Assouline & Or, 2013). Leaf gas exchange models often express diffusive resistance as a function of stomata pore size only and not of stomata spacing. By defining the diffusion through the vapor shell as a property of the stoma, the end-correction resistance Rend and the resistance of the stomatal pore depth RSP are combined and lumped into the leaf resistance RLeaf (or its inverse, the leaf conductance: gLeaf = 1/RLeaf). The resulting term expresses the combined resistances of the pore depth extended by a length x (end-correction), according to: RLeaf
1 d þx pr ¼ ¼ RSP þ Rend ! x ¼ n k p r2 4
Eqn 4
Note that for Rend we inserted here Eqn 2(a). Various formulations of ‘end-correction’ are found in literature; to Ó 2015 The Authors New Phytologist Ó 2015 New Phytologist Trust
facilitate a systematic comparison, we define x = cendr with endcorrection factor cend. The most frequently used values for cend are: cend = p/4 applied by Franks & Farquhar (2001) corresponding to one-sided resistance Rend as defined by Stefan (1882); or cend = p/2 related to a resistance Rend at both ends of stomatal pore, as used in Franks & Beerling (2009), Ting & Loomis (1965), de Boer et al. (2011) and Dow et al. (2014). Nobel (2005) proposed an intermediate value: cend = 1, indicating that the resistance related to vapor diffusion at the ends of the stomatal pore may be smaller than expected from two end-corrections. To account for the limited spacing between stomata (on the leaf epidermis) we employ in the following the gas diffusive resistance model of Bange (1953) as in Eqn 2(c) for interacting vapor shells. For the sub-stomatal cavity side, we assign for simplicity (and due to lack of specific information on geometry of sub-stomatal cavity) a single end-correction according to Eqn 2(a). The last component of diffusive resistance considered is between the vapor shell and the top of an attached air boundary layer of thickness d that decreases with increasing mean wind speed. This diffusive resistance designated as RBL (the index ‘BL’ for boundary layer) could also be used to define the potential evaporation rate from free water surfaces, expressed as: 1 k ¼ gBL ¼ RBL d
Eqn 5
Stomata spacing effects on diffusive fluxes The total diffusive resistance is computed in a manner analogous to estimating resistance in an electrical circuit, with resistances arranged in series or in parallel (Weyers & Meidner, 1990). The total diffusive resistance is typically computed per total leaf area with n stomata arranged in parallel (with conductances gend, gSP and gVS that are proportional to stomatal density n), and arranging this resistance with the boundary layer resistance RBL = d/k in series. To better understand conceptually how stomata density (and spacing) affect diffusive fluxes, we analyze diffusive resistance for a unit area around a single stoma (for simplicity, a pffiffiffi square with size s ¼ 1= n ) with resistance ℜ [T mol1] assigned to the unit area (with symbol ℜ to differentiate from resistance R [T L2 mol1] assigned to a total leaf area). For each stoma the total resistance is the sum of ℜend (i.e. the end-correction for the sub-stomatal entry), the resistance ℜSP in the stoma pore, the resistance of the vapor shell ℜVS that could be limited by spacing s, and the boundary layer resistance ℜBL. These resistances are expressed as follows:
Effects of stomata clustering on leaf gas exchange Peter Lehmann* and Dani Or* Soil and Terrestrial Environmental Physics, ETH Zurich, Universit€atstrasse 16, 8092 Zurich, Switzerland
Summary Author for correspondence: Peter Lehmann Tel: +41 44 632 63 45 Email: [email protected] Received: 7 October 2014 Accepted: 17 March 2015
New Phytologist (2015) 207: 1015–1025 doi: 10.1111/nph.13442
Key words: gas diffusion, leaf conductance, spatial organization, stomata clustering, stomatal aperture.
A general theoretical framework for quantifying the stomatal clustering effects on leaf gaseous diffusive conductance was developed and tested. The theory accounts for stomatal spacing and interactions among ‘gaseous concentration shells’. The theory was tested using the unique measurements of Dow et al. (2014) that have shown lower leaf diffusive conductance for a genotype of Arabidopsis thaliana with clustered stomata relative to uniformly distributed stomata of similar size and density. The model accounts for gaseous diffusion: through stomatal pores; via concentration shells forming at pore apertures that vary with stomata spacing and are thus altered by clustering; and across the adjacent air boundary layer. Analytical approximations were derived and validated using a numerical model for 3D diffusion equation. Stomata clustering increases the interactions among concentration shells resulting in larger diffusive resistance that may reduce fluxes by 5–15%. A similar reduction in conductance was found for clusters formed by networks of veins. The study resolves ambiguities found in the literature concerning stomata end-corrections and stomatal shape, and provides a new stomata density threshold for diffusive interactions of overlapping vapor shells. The predicted reduction in gaseous exchange due to clustering, suggests that guard cell function is impaired, limiting stomatal aperture opening.
Introduction Studies have shown that leaf stomatal size and density evolved over geological timescales; these structural changes are interpreted (using gas diffusion theories) as refinements in plant regulation of gas exchange (Raven, 2002; Franks & Beerling, 2009; Berry et al., 2010; de Boer et al., 2011; Assouline & Or, 2013; Buckley & Schymanski, 2013). Stomata patterns on leaves have received much less attention than the dynamic function of stomata (Croxdale, 2000). Various studies have revealed that an uniform stomata pattern is linked to the ‘one cell spacing rule’ that defines a minimum spacing between adjacent stomata, often of several stomata aperture diameters (Korn, 1993; Larkin et al., 1997; Geisler et al., 2000). This ‘rule’ ensures that there are sufficient intervening cells for the proper function of guard cells whilst reducing the diffusive interactions among neighboring stomata (Bange, 1953; Franks & Casson, 2014). However, as already observed by Fellerer (1892) for Begonia, various plant species produce stomata arranged in clusters (Gan et al., 2010; listed 39 families of land plants). Stomata clustering may result from errors in the complex signaling for stomata initiation and in cell orientation and division (Yang & Sack, 1995; Bergmann & Sack, 2007; Peterson et al., 2010; Akita et al., 2013). Environmental signals may also regulate stomata development and patterning (Wang et al., 2007), such as signals from older to younger leaves to reduce stomata density in water-stressed plants (Serna & Fenoll, 1997), or the formation of *These authors contributed equally to this work. Ó 2015 The Authors New Phytologist Ó 2015 New Phytologist Trust
stomata clusters in certain plants grown in dry environments (Hoover, 1986). The formation of stomata clusters was posited to reflect an adaption to dry environments aimed at reducing water loss from plant leaves (Gan et al., 2010; Franks & Casson, 2014). Recently, Dow et al. (2014) measured the effect of stomatal clustering on maximum leaf conductance of various genotypes of Arabidopsis thaliana expressing different stomatal patterns. Their results show that diffusive conductance from leaves with clustered stomata was 60% lower relative to those with uniformly distributed stomata for densities (sum of abaxial and adaxial leaf surface) in the range of 300–450 stomata per mm2. Dow et al. (2014) hypothesized that the observed reduction is probably due to impaired function of the tightly spaced stomata guard cells that may prevent complete opening of the stomata. However, an additional potential effect of clustering may involve increased diffusive resistance within stomata clusters as postulated by Gan et al. (2010) and Franks & Casson (2014), and recently shown theoretically and experimentally for other evaporating porous surfaces by Lehmann & Or (2013). To quantify the potential effects of stomatal clustering on gaseous diffusive resistance, we developed a physically based analytical model that accounts for the main components of gas exchange resistance in distributed and clustered stomata patterns on plant leaves. The expressions developed provide a theoretical basis for predicting gaseous flux reduction due to different stomatal clustering scenarios, and indirectly provide a proper context for the physiological impairment of guard cell hypothesis. In Supporting Information Notes S1 the model is applied to quantify gas conductance for a New Phytologist (2015) 207: 1015–1025 1015 www.newphytologist.com
New Phytologist
1016 Research
different type of clustering defined by the network of veins, indicating that the conduction is reduced by this type of clustering as well.
Materials and Methods Theory of diffusive gas transport through stomata The diffusive gas flux F (expressed as moles per unit area per unit time) through stomata-perforated leaves depends on the driving force (i.e. a concentration difference along the diffusive pathway), and on the conductance g [mol L2 T1] of the various obstructions between sub-stomatal cavity and the ambient atmosphere. It is convenient to use diffusive resistance R (i.e. the reciprocal of conductance g), to express the total resistance as a series of different resistances as presented in Fig. 1. In the following, we quantify the resistance and conductance of these elements and their dependence on stomata size, shape and spatial arrangement. Note that we often discuss the various terms in relation to ‘vapor transport’ and transpiration, but the same expressions apply to CO2 uptake as well. Computing leaf conductance for gas transport We first consider gaseous diffusive conductance of a stomatal pore gSP (marked by index SP for ‘stomatal pore’) of depth d [L] and cross-sectional area A [L2], defined by Brown & Escombe (1900) as: 1 A D p r2 k n ¼ gSP ¼ n ¼ RSP d v d
Eqn 1
with presistance RSP, pore radius r [L] (same area A with ffiffiffiffiffiffiffiffiffi r ¼ A=p), stomatal density n [L2] and the ratio k [mol L1 T1] between vapor diffusion coefficient D [L2 T1]
and molar volume of air v [L3 mol1] (with k ~ 103 mol s1 m1 for water). Brown & Escombe (1900) considered an additional diffusive resistance at the ends of a pore related to converging and diverging concentration shells forming at pore apertures. Stomata spacing plays a role in limiting the lateral extent of adjacent vapor shells. We distinguish between the standard gas exchange formulation, which defines this resistance as a property of the stoma itself (the ‘end-correction’), and formulations that consider the effects of stomata spacing on diffusive resistance across ‘concentration shells’ (e.g. Bange, 1953; Schl€ under, 1988). This raises a certain ambiguity present in the literature concerning the proper representation of diffusive resistance related to ‘end-correction’, the role of stomata shape, and stomata spacing. Specifically, we wish to clarify (1) the number of end-corrections to be considered (at one or both ends of the stomata pore), the effects of (2) stomata shape (circular or elliptic), and the influence of (3) spacing s between adjacent stomata. In Fig. S2 we show how assumptions concerning stomata shape may lead to incorrect conclusions regarding the end-correction. In the following we summarize the various expressions used in the literature to quantify the ‘concentration shell’ resistance (i.e. for one concentration shell; for concentration shells at both sides the expressions must be doubled). Note that because we will discuss the transport of vapor (transpiration) in detail we will use in the following the specific term ‘vapor shell’ instead of the more general term ‘gaseous concentration shell’. We differentiate between approaches that consider the limited spacing and interaction between vapor shells (marked by index ‘VS’ for interacting vapor shells) and end-correction formulations that do not consider the spacing of the vapor shells (with index ‘end’ for end-correction). End-corrections (no interaction) Rend ¼
RBL
Flux per stoma
δ
s
RSP Rend
RLeaf
2r d
Fig. 1 Resistances for vapor diffusion through stomata-perforated leaves with resistance through the stomatal pore RSP, vapor shells forming at the end of each stomatal aperture RVS that may interact with each other (related to end-correction Rend, see ‘Discussion’ in the text), and boundary layer resistance RBL. The resistances of the vapor shells and the pore resistance are combined and lumped into leaf resistance RLeaf. The total resistance of leaf and atmosphere in series is controlled by four dimensions: stomatal depth d, pore radius r, spacing between stomata s and thickness of boundary layer d. New Phytologist (2015) 207: 1015–1025 www.newphytologist.com
Eqn 2(a)
(Brown & Escombe, 1900)
Rend ¼
RVS
1 1 ¼ gend 4r k n
Loge ð4a=bÞ 1 ¼ gend 2p a k n
Eqn 2(b)
(Parlange & Waggoner, 1970) Interacting vapor shells
RVS ¼
1 ¼ gVS
1 1 1 4r p s k n
Eqn 2(c)
(Bange, 1953) Ó 2015 The Authors New Phytologist Ó 2015 New Phytologist Trust
New Phytologist RVS
1 ¼ ¼ gVS
Research 1017
1 1 1 2p r p s k n
Eqn 2(d)
(Schl€ under, 1988) with spacing s between stomata and radius of major and minor axis of ellipsoidal-shaped stomata, a and b, respectively. Eqn 2(a) for the resistance of a vapor shell forming above a circular pore was proposed by Stefan (1882) and subsequently used by Brown & Escombe (1900); for elliptical stomatal pores with b a, Eqn 2(b) from Parlange & Waggoner (1970) is more accurate (see discussion in Notes S2). The equations derived by Bange (1953) and Schl€ under (1988) consider the effect of limited spacing between pores on the diffusive resistance through the vapor shells. The derivations of Bange and Schl€ under differ with respect to the shape of the flux source; this is a disk in Eqn 2(c) from Bange (1953), and a hemisphere in Eqn 2(d) from Schl€ under (1988). Interestingly, Eqn 2(c) converges to Brown and Escombe’s end-correction in Eqn 2(a) for large stomata spacing s. Rend and diffusive resistance of interacting vapor shells RVS can be related (Eqn 3a) to define a critical density nVS (Eqn 3b) where the effect of vapor (or gas) shell interactions becomes negligible (< 10% of the end-correction resistance): pffiffiffi RVS 4r 4r n ¼1 ¼1 ps Rend p
Eqn 3(a)
p 2 RVS 1 0:9 ! nVS 40r 160 r 2 Rend
Eqn 3(b)
for stomata pffiffiffi spacing s defined on a square grid with size s ¼ 1= n. For small stomata densities n (and small pore radius r), the second term on the RHS of Eqn 3(a) becomes small; hence, the gas diffusion model of Bange (1953) that considers interacting vapor shells converge to standard end-correction formulation. Details of the differences between the two expressions are discussed in the Notes S3 and explained in specific examples shown in Fig. S3. With a few exceptions, the effects of stomatal spacing s on diffusive conductance through interacting vapor shells forming around a stomatal pore is rarely considered (Bange, 1953; Assouline & Or, 2013). Leaf gas exchange models often express diffusive resistance as a function of stomata pore size only and not of stomata spacing. By defining the diffusion through the vapor shell as a property of the stoma, the end-correction resistance Rend and the resistance of the stomatal pore depth RSP are combined and lumped into the leaf resistance RLeaf (or its inverse, the leaf conductance: gLeaf = 1/RLeaf). The resulting term expresses the combined resistances of the pore depth extended by a length x (end-correction), according to: RLeaf
1 d þx pr ¼ ¼ RSP þ Rend ! x ¼ n k p r2 4
Eqn 4
Note that for Rend we inserted here Eqn 2(a). Various formulations of ‘end-correction’ are found in literature; to Ó 2015 The Authors New Phytologist Ó 2015 New Phytologist Trust
facilitate a systematic comparison, we define x = cendr with endcorrection factor cend. The most frequently used values for cend are: cend = p/4 applied by Franks & Farquhar (2001) corresponding to one-sided resistance Rend as defined by Stefan (1882); or cend = p/2 related to a resistance Rend at both ends of stomatal pore, as used in Franks & Beerling (2009), Ting & Loomis (1965), de Boer et al. (2011) and Dow et al. (2014). Nobel (2005) proposed an intermediate value: cend = 1, indicating that the resistance related to vapor diffusion at the ends of the stomatal pore may be smaller than expected from two end-corrections. To account for the limited spacing between stomata (on the leaf epidermis) we employ in the following the gas diffusive resistance model of Bange (1953) as in Eqn 2(c) for interacting vapor shells. For the sub-stomatal cavity side, we assign for simplicity (and due to lack of specific information on geometry of sub-stomatal cavity) a single end-correction according to Eqn 2(a). The last component of diffusive resistance considered is between the vapor shell and the top of an attached air boundary layer of thickness d that decreases with increasing mean wind speed. This diffusive resistance designated as RBL (the index ‘BL’ for boundary layer) could also be used to define the potential evaporation rate from free water surfaces, expressed as: 1 k ¼ gBL ¼ RBL d
Eqn 5
Stomata spacing effects on diffusive fluxes The total diffusive resistance is computed in a manner analogous to estimating resistance in an electrical circuit, with resistances arranged in series or in parallel (Weyers & Meidner, 1990). The total diffusive resistance is typically computed per total leaf area with n stomata arranged in parallel (with conductances gend, gSP and gVS that are proportional to stomatal density n), and arranging this resistance with the boundary layer resistance RBL = d/k in series. To better understand conceptually how stomata density (and spacing) affect diffusive fluxes, we analyze diffusive resistance for a unit area around a single stoma (for simplicity, a pffiffiffi square with size s ¼ 1= n ) with resistance ℜ [T mol1] assigned to the unit area (with symbol ℜ to differentiate from resistance R [T L2 mol1] assigned to a total leaf area). For each stoma the total resistance is the sum of ℜend (i.e. the end-correction for the sub-stomatal entry), the resistance ℜSP in the stoma pore, the resistance of the vapor shell ℜVS that could be limited by spacing s, and the boundary layer resistance ℜBL. These resistances are expressed as follows:
