We develop a model to approximate the physiology of C3 angiosperm species with abaxial stomata. C3 species dominate terrestrial ecosystem productivity, comprise the largest fraction of contemporary plant diversity, as well as the paleodiversity after the late-Mesozoic angiosperm radiation (Stewart & Rothwell, 1993). In principle, the model could be expanded to include the specific anatomies and physiologies of other groups (e.g. gymnosperms, ferns, bryophytes or monocots). All model parameters are summarized in Table 1.
Table 1. Summary of model parameters
| T c ||Growing season temperature||°C||10||5|
| C a ||Atmospheric carbon dioxide pressure||Pa||40||10|
| h ||Atmospheric humidity||%||60||10|
| D ||Vapor pressure deficit||kPa||1||0.5|
| g 1 ||Stomatal conductance coefficient||–||3||2|
|ΔΨls||Leaf–stem water potential||MPa||0.10||0.05|
| d y ||Leaf half-thickness||μm||80||30|
| s ||Insolation factor||–||1||0.1|
A key assumption is that, under ‘ideal conditions’, leaf water supply (E; transpiration rate; mmol m−2 s−1) is proportional (α; dimensionless) to environmental water demand (PET; potential evapotranspiration; mmol m−2 s−1) where:
Here, ‘ideal conditions’ means that the model will be most valid when leaves are functioning at low levels of physiological stress (e.g. open stomata and favorable leaf–stem water potentials). As a result, the venation network, which is constructed early during leaf development (Sack et al., 2012), should have a structure that matches, but does not exceed, the environmental demands that the leaf is likely to experience over a typical lifespan. This hypothesis is consistent with the coordination of leaf hydraulics with environmental conditions in several species (Brodribb & Jordan, 2011; Carins Murphy et al., 2012, 2014; Blonder et al., 2013). Moreover, variation in leaf minor vein density matches variation in environmental water demand (Uhl & Mosbrugger, 1999; Givnish et al., 2005; Sack & Frole, 2006; Brodribb & Feild, 2010; Brodribb & Jordan, 2011) at both the developmental and evolutionary timescales (but see Feild et al., 2011).
Next, we assume that leaf transpiration rate, E, will be related to the leaf–stem water potential, ΔΨls (MPa); and the leaf hydraulic path length dm (μm). We formally link E, ΔΨls and dm using an empirical relationship reported by Brodribb et al. (2007):
Further, we can write dm, the hydraulic path length (μm), as
where dy is the leaf half-thickness (μm), representing the characteristic distance from vein to abaxial stomata, and div is the distance between minor veins, which is empirically related to vein density (VD; mm−1) as
Note that this transpiration model may lose some accuracy depending on the mode of water transport but deviations are important only at very high VD values (de Boer et al., 2012).
The proportionality factor α empirically relates actual evapotranspiration to potential evapotranspiration (Trabucco & Zomer, 2010). This coefficient is used in the context of agricultural crop coefficients (Allen et al., 1998) and is similar to the Horton index, which is defined as the ratio of water vaporization through any means to catchment wetting at the landscape scale (Troch et al., 2009). We define α as the Priestley–Taylor coefficient, the fraction of surface moisture available for evaporation (Priestley & Taylor, 1972), because it can be empirically linked to stomatal conductance (eqn (3) of Komatsu, 2005):
(gs, stomatal conductance to water vapor (mm s−1)).
We assume that, across differing climates, leaves maximize carbon gain per unit water loss by regulating stomatal conductance (Cowan & Farquhar, 1977). Based on a recent optimality model (Medlyn et al., 2011) we use:
where 10−3 RT0 is a unit conversion factor from mol m−2 s−1 to mm s−1, R = 8.31447 J mol−1 K−1 and T0 = 288.15 K, 1.6 converts from conductance of CO2 to that of H2O (A, the peak photosynthetic rate (μmol m−2 s−1); Ca, atmospheric CO2 pressure (Pa); g1, a constant (kPa1/2) reflecting the marginal water cost of carbon and the CO2 compensation point for photosynthesis; D, vapor pressure deficit (kPa)). For analytic tractability, we linearize Eqn (6) in terms of Ca by performing a first-order Taylor approximation around a central value, Ca*:
here choosing Ca* = 40 (the modern atmospheric value).
We make a further assumption that venation networks are coupled to photosynthesis (Brodribb et al., 2007). Primarily for analytic tractability, we make the linear approximation that
We use η = 1 (mm μmol m−2 s−1) based on an approximate fit to the data of (Brodribb et al., 2007).
PET is modeled using a modified version of the Hargreaves–Samani (Hargreaves & Samani, 1982) model. This version was chosen because it requires relatively few parameters:
where the 625/972 prefactor converts from mm d−1 to mmol m−2 s−1 (Ra, total incident solar radiation; Ct, a humidity factor; Tf, temperature (°F); ΔT, average temperature range). We convert Tf to a Celsius temperature (Tc; °C) as Tf = 1.8 · Tc + 32, where again, Tc is the growing season temperature, with Ra defined as
and where s is an insolation factor (dimensionless), and orbital parameters dr, ωs and δ are given as
( J, day of year (dimensionless); θ, latitude (°)). The humidity factor is empirically defined (Hargreaves & Samani, 1982) as
where h is the relative humidity (%). The isothermality factor ΔT can be empirically written (Hargreaves & Samani, 1982) as
In order to reduce the number of free parameters in the model, we use the assumption that the model applies only during the ‘ideal’ conditions previously described. Under these conditions, we can further parameterize the model with J = 180 (midsummer), D = 1 and g1 = 3 (typical for the species being modeled) (Medlyn et al., 2011) and ΔΨls = 0.10 (typical under low water-stress conditions across diverse species) (Sack & Holbrook, 2006).
Lastly, we assume that leaf thickness is a constant. This simplifying assumption reduces the dimensionality of the model and makes it possible to use the model for paleoclimate applications. Thickness cannot be easily measured on fossil leaves. We choose dy = 80 because it represents many common species and is commonly used in other paleoecological models (de Boer et al., 2012). Modern insolation intensities are by definition characterized by s = 1. Across paleotime, the value of s may be variable, depending on orbital variation, solar output (Laskar et al., 2011) and mean cloudiness.
We solve the above equations analytically to predict atmospheric carbon dioxide Ca (or Tc) based on measured values of leaf venation, VD, the growing season temperature Tc (or Ca), latitude θ and relative humidity h. While the solutions are too large to present here, they are shown in full in the Supporting Information (Eqns S1–S3 in Notes 1). We also provide a Mathematica notebook (Notes S2) for direct manipulation and parameterization of the model. In general, higher values of VD are predicted to yield higher values of Tc or lower values of Ca, if all other parameters are held constant. The analytic form of these equations also makes it possible to explore the consequences of variation in traits that we assumed to be constant (e.g. leaf thickness, water potential).
We tested predictions of our model by collecting leaves from species found within 17 sites along a temperate and tropical elevation gradient (Table 2). At each site we measured latitude and longitude using a GPS unit. For tropical sites, elevation was obtained directly from satellites; for temperate sites, elevation was obtained from a digital elevation model (USGS, National Elevation Dataset).
Table 2. Summary of collections at each location (the number of leaves collected follows each species name)
|65||Annonaceae/Oxandra venezuelana (1), Apocynaceae/Aspidosperma desmanthum (3), Bignoniaceae/Arrabidaea sp.2 (1), Bignoniaceae/Callichlamys latifolia (1), Boraginaceae/Cordia sp.2 (1), Chrysobalanaceae/Licania operculipetala (1), Connaraceae/Rourea glabra (1), Erythroxylaceae/Erythroxylum macrophyllum (3), Fabaceae/Clitoria javitensis (2), Fabaceae/Machaerium kegelii (1), Fabaceae/Machaerium salvadorense (1), Fabaceae/Swartzia ochnacea (1), Icacinaceae/Discophora guianensis (1), Lacistemataceae/Lacistema aggregatum (2), Lauraceae/Nectandra umbrosa (3), Lauraceae/Ocotea leucoxylon (1), Melastomataceae/Mouriri gleasoniana (2), Meliaceae/Ruagea glabra (2), Moraceae/Trophis involucrata (1), Myrtaceae/Eugenia acapulcensis (1), Myrtaceae/Myrciaria floribunda (1), Rubiaceae/Faramea occidentalis (1), Salicaceae/Tetrathylacium johansenii (1), Sapotaceae/Pouteria chiricana (3)|
|500||Brassicaceae/Capparis frondosa (1), Burseraceae/Protium glabrum (1), Burseraceae/Protium sp.1 (1), Clusiaceae/Garcinia intermedia (1), Clusiaceae/Tovomita longifolia (1), Fabaceae/Inga coruscans (1), Fabaceae/Macrolobium costaricense (1), Fabaceae/Swartzia ochnacea (1), Lauraceae/Nectandra umbrosa (1), Moraceae/Clarisia biflora (1), Moraceae/Pseudolmedia glabrata (1), Moraceae/Sorocea hispidula (1), Myristicaceae/Compsoneura excelsa (3), Myristicaceae/Virola guatemalensis (1), Myrtaceae/Eugenia acapulcensis (2), Rubiaceae/Chione venosa (1), Salicaceae/Lunania mexicana (1), Salicaceae/Tetrathylacium johansenii (1), Sapotaceae/Chrysophyllum sp.2 (1), Violaceae/Rinorea crenata (8), Violaceae/Rinorea hummelii (1), Violaceae/Rinorea squamata (2), Vochysiaceae/Vochysia megalophylla (1)|
|1050||Annonaceae/Guatteria diospyroides (1), Apocynaceae/Lacmellea zamorae (2), Burseraceae/Protium sp.3 (1), Burseraceae/Protium sp.4 (1), Dichapetalaceae/Dichapetalum sp.1 (1), Euphorbiaceae/Croton megistocarpus (1), Euphorbiaceae/Hieronyma oblonga (1), Euphorbiaceae/Richeria dressleri (2), Fabaceae/Entada gigas (1), Fabaceae/Inga latipes (1), Fabaceae/Inga thibaudiana (1), Icacinaceae/Discophora guianensis (1), Lauraceae/Licaria sp.1 (1), Lauraceae/Ocotea meziana (1), Lauraceae/Ocotea praetermissa (1), Moraceae/Brosimum guianense (2), Moraceae/Pseudolmedia glabrata (1), Moraceae/Pseudolmedia mollis (1), Myrsinaceae/Ardisia dunlapiana (2), Myrtaceae/Myrcia sp.2 (1), Rubiaceae/Coussarea caroliana (2), Rubiaceae/Coussarea loftonii (2), Rubiaceae/Faramea sp.1 (1), Rubiaceae/Posoqueria coriacea (1), Rubiaceae/unknown sp.1 (1), Sapindaceae/Matayba apetala (1), Sapotaceae/Chrysophyllum sp.2 (1), Sapotaceae/Chrysophyllum sp.3 (1), Sapotaceae/Pouteria chiricana (1), Verbenaceae/Aegiphila sp.1 (1), Vochysiaceae/Vochysia allenii (1)|
|2050||Annonaceae/Guatteria oliviformis (1), Aquifoliaceae/Ilex skutchii (1), Aquifoliaceae/Ilex sp.2 (1), Araliaceae/Dendropanax querceti (2), Araliaceae/Oreopanax xalapensis (6), Asteraceae/Verbesina oerstediana (4), Bignoniaceae/Amphitecna sessilifolia (1), Brunelliaceae/Brunellia costaricensis (2), Fagaceae/Quercus copeyensis (1), Fagaceae/Quercus rapurahuensis (1), Fagaceae/Quercus seemannii (2), Lauraceae/Ocotea insularis (5), Lauraceae/Ocotea praetermissa (1), Lauraceae/Ocotea sp.1 (1), Lauraceae/Ocotea valeriana (1), Magnoliaceae/Magnolia poasana (1), Myrsinaceae/Ardisia sp.3 (1), Myrsinaceae/unknown sp.2 (2), Rubiaceae/Palicourea sp.2 (1), Rubiaceae/Rondeletia amoena (1), Rubiaceae/Rondeletia buddleioides (2), Sabiaceae/Meliosma vernicosa (5)|
|2430||Aquifoliaceae/Ilex sp.2 (1), Aquifoliaceae/Ilex sp.3 (1), Araliaceae/Dendropanax querceti (1), Asteraceae/Verbesina oerstediana (1), Brunelliaceae/Brunellia costaricensis (1), Caprifoliaceae/Viburnum stellatotomentosum (1), Chloranthaceae/Hedyosmum goudotianum (2), Cornaceae/Cornus disciflora (2), Cunoniaceae/Weinmannia pinnata (2), Ericaceae/Satyria warszewiczii (1), Ericaceae/Vaccinium consanguineum (2), Fagaceae/Quercus copeyensis (3), Fagaceae/Quercus rapurahuensis (1), Fagaceae/Quercus seemannii (1), Juglandaceae/Alfaroa costaricensis (3), Lauraceae/Nectandra cufodontisii (1), Lauraceae/Ocotea pittieri (3), Magnoliaceae/Magnolia sororum (1), Malpighiaceae/Bunchosia ternata (2), Melastomataceae/Miconia sp.1 (1), Meliaceae/Trichilia havanensis (3), Myrsinaceae/Ardisia glandulosomarginata (3), Myrsinaceae/Myrsine juergensenii (3), Rubiaceae/Palicourea salicifolia (1), Rutaceae/Zanthoxylum melanostictum (1), Styracaceae/Styrax argenteus (1), Symplocaceae/Symplocos retusa (1)|
|3250||Araliaceae/Oreopanax xalapensis (1), Araliaceae/Schefflera rodriguesiana (2), Asteraceae/Diplostephium costaricense (2), Celastraceae/Maytenus woodsonii (2), Clethraceae/Clethra gelida (2), Clusiaceae/Hypericum irazuense (1), Cunoniaceae/Weinmannia pinnata (3), Ericaceae/Comarostaphylis arbutoides (2), Ericaceae/Macleania rupestris (2), Ericaceae/Vaccinium consanguineum (3), Escalloniaceae/Escallonia myrtilloides (1), Fagaceae/Quercus costaricensis (1), Garryaceae/Garrya laurifolia (3), Melastomataceae/Miconia schnellii (1), Melastomataceae/Miconia talamancensis (2), Myrsinaceae/Myrsine dependens (2), Rhamnaceae/Rhamnus oreodendron (3), Scrophulariaceae/Buddleja nitida (1)|
|2437||Malvaceae/Sphaeralcea coccinea (9), Rosaceae/Prunus emarginata (9), Rosaceae/Prunus virginiana (9), Rosaceae/Rosa acicularis (9), Salicaceae/Populus angustifolia (9)|
|2481||Asteraceae/Artemisia tridentata (9), Asteraceae/Balsamorhiza sagittata (9), Asteraceae/Psilochenia occidentalis (9), Rosaceae/Amelanchier utahensis (9), Scrophulariaceae/Castilleja chromosa (9)|
|2706||Asteraceae/Chrysanthemum leucanthemum (9), Asteraceae/Senecio integerrimus (9), Asteraceae/Taraxacum officinale (9), Brassicaceae/Thlaspi montanum (9), Fabaceae/Lupinus bakeri (9), Ranunculaceae/Delphinium nuttalianum (9), Rosaceae/Geum triflorum (9), Rosaceae/Pentaphylloides floribunda (9), Salicaceae/Salix wolfii (3), Valerianaceae/Valeriana edulis (9), Valerianaceae/Valeriana occidentalis (9)|
|2807||Asteraceae/Achillea millefolia (9), Asteraceae/Chrysothamnus parryi (9), Asteraceae/Erigeron speciosus (9), Asteraceae/Hymenoxys hoopesii (9), Fabaceae/Lathyrus lanzwertii (9), Fabaceae/Vicia americana (9), Geraniaceae/Geranium viscossimum (9), Polygonaceae/Erigonum umbellatum (9), Rosaceae/Potentilla gracilis (9), Salicaceae/Salix sp.1 (9), Salicaceae/Salix sp.2 (9), Violaceae/Viola sororia (9)|
|2873||Asteraceae/Artemisia frigida (9), Fabaceae/Trifolium repens (9), Onagraceae/Epilobium angustifolium (9)|
|2889||Berberidaceae/Mahonia repens (9), Polemoniaceae/Polemonium foliosissimum (9), Rosaceae/Rubus idaeus (9)|
|2921||Adoxaceae/Sambucus racemosa (9), Asteraceae/Arnica cordifolia (9), Orobanchaceae/Pedicularis bracteosa (9), Rosaceae/Fragaria virginiana (9)|
|2922||Caprifoliaceae/Loniceria involucrata (9), Celastraceae/Paxistima myrsinites (9)|
|2940||Apiaceae/Osmorhiza occidentalis (9), Geraniaceae/Geranium richardsonii (9), Ranunculaceae/Thalictrum fendleri (9), Rubiaceae/Galium septentrional (9), Salicaceae/Populus tremuloides (9)|
|3162||Apiaceae/Heracleum sphondylium (9), Boraginaceae/Mertensia fusiformis (9), Crassulaceae/Sedum rosea (9), Onagraceae/Epilobium angustifolium (9)|
|3171||Apiaceae/Ligusticum tenuifolium (9), Boraginaceae/Hydrophyllum capitatum (9), Grossulariaceae/Ribes montigenum (9), Ranunculaceae/Aquilegia coerulea (9)|
|3357||Gentianaceae/Frasera speciosa (9), Papaveraceae/Corydalis caseana (9)|
|3368||Asteraceae/Erigeron sp.2 (9), Asteraceae/Heterotheca villosa (9), Gentianaceae/Gentiana affinis (9), Polygonaceae/Rumex densiflorus (9)|
The temperate gradient was located in the Gunnison Valley of western Colorado in the United States (39°N) and included 11 sites (each c. 1 m2) ranging in elevation from c. 2440 to 3370 m asl. These sites span a continuum from arid montane riparian areas to alpine meadow and include both woody and herbaceous species. During the 2010 growing season we sampled the more common nonmonocot angiosperm species (n = 6 ± 3 SD), taking several (9 ± 1 SD) mature undamaged leaves from individuals of each species.
The tropical gradient was located in the Savegre River drainage of western Costa Rica (9°N) and included six sites ranging in elevation from 65 to 3250 m. These sites span a continuum from tropical moist forest to tropical wet montane forest and include only woody species. For each site we set up a 0.1-ha ‘Gentry transect’ (Phillips & Miller, 2002), identifying every individual with dbh ≥ 2.5 cm. We then sampled at least one leaf from at least one individual of every observed species. At each site, we then chose a random subset of fifty leaves, each from a different individual for venation analysis. We took this random sampling approach because the high diversity at these tropical sites (mean richness = 68 ± 31 SD species) made a full analysis of all leaves time-prohibitive.
Vein density measurements
All leaves were pressed flat and dried at 60°C for at least 3 d. We then cleared each leaf to expose its venation using established protocols (Pérez-Harguindeguy et al., 2013). We cut a 1-cm2 section from each leaf, selecting a region of the lamina that did not include any primary veins. We immersed the leaf sample in a solution of 5% w/v sodium hydroxide : water heated to a temperature of 50°C for up to 7 d, until the leaf became transparent. We then rinsed the leaf in water and transferred it to a 2.5% w/v sodium hypochlorite : water solution for up to 5 min, until the leaf became white. We then rinsed the leaf in water and transferred it to 50% v/v ethanol : water solution for 5 min, and then to a staining solution of 0.1% w/v safranin : ethanol for 30 min. We then transferred the leaf to a destaining solution of 100% ethanol for 1 h before transferring to 50% v/v ethanol : toluene for 30 s and then to 100% toluene. We then mounted each leaf on a glass slide using the toluene-based Permount medium (Fisher Scientific, Waltham, MA, USA). We allowed slides to dry for 3 d during which the clearing process continued. Some samples were inadvertently destroyed by this chemical process. The final dataset included 225 tropical leaves and 529 temperate leaves from 186 nonmonocot angiosperm species.
We then imaged each leaf using a dissecting microscope (SZX-12; Olympus) coupled to a digital camera (T2i; Canon, Japan). Slides were back-illuminated using a light box. Images were obtained at a final resolution of 430 pixels per millimeter with a full extent of c. 10 mm × 7 mm. We then retained only the green channel of each image and applied a contrast-limited adaptive histogram equalization procedure to improve image quality.
We estimated vein density on each image using a stochastic line-intersection technique. The distance between veins is known to strongly correlate with the density of veins (Uhl & Mosbrugger, 1999; Brodribb et al., 2007). Distance can be rapidly estimated by counting the number of veins crossed by a line of a known length (cartooned in Fig. S1). To calibrate this approach, we first used a collection of previously traced leaves from 25 morphologically diverse species (Blonder et al., 2011) on which we simulated the placement of a number of randomly oriented line segments. We then compared the known vein density of the leaf to the mean distance between veins, as estimated as the total length of all line segments divided by the total number of vein intersections.
For as few as 10 random line segments (c. 7 cm total length) there was a very strong correlation (r2 = 0.89, P < 10−15) between vein density (VD, mm−1) and distance (d, mm):
We then pooled leaf-level measurements to calculate species-at-site mean vein densities and used these to then estimate site-mean vein density. We used species-at-site means because some species occurred at multiple sites, potentially obscuring trait variation due to between-site climate variation.
Model uncertainty analysis
We measured the impact on Tc and Ca of two classes of uncertainty in the model: sampling error in VD and systemic error in all other model parameters. We first solved the model analytically for Tc and Ca. To assess systemic error in all model parameters, we assumed that the remaining parameters (D, ΔΨls, g1, dy, s) were random variables uniformly distributed with a central value and a half-width reflecting to a physiologically relevant range (Table 1). When solving for Tc, we assumed that Ca was uniformly distributed between 30 and 50 Pa; when solving for Ca, we assumed that Tc was uniformly distributed between 5 and 15°C. We also allowed latitude to vary 1° in half-width around the observed value. To assess measurement error in VD, we assumed that VD was uniformly distributed between the 25% and 75% quantile of its distribution at each site. We sampled parameter values from each distribution and calculated the resulting Tc (or Ca) value.
We obtained parameter deviations by subtracting these parameter values from their central values, and prediction deviations by subtracting the Tc (or Ca) value from the value predicted when using central values for all parameters. Next, we repeated the resampling 1000 times per analysis. We calculated the middle quartile of each predicted deviation as a combined uncertainty estimate for each site. We also directly measured sampling uncertainty by solving the model, holding all parameters constant to their central values except VD, which was allowed to vary as above. All deviations are reported as interquartile ranges of these distributions.
We also measured the relative importance of each parameter to predictions of Ca (or Tc). We repeated the sensitivity analysis with the above parameter distributions, this time assuming VD to be uniformly distributed across its global range (c. 1–25 mm−1; Boyce et al., 2009). We then constructed a linear model for deviations in Ca (or Tc) as a function of deviations in all parameters. For each parameter, we report the effect direction as the sign of the regression coefficient; and the overall effect size as the ratio of the parameter's explained sum of squares divided by the total sum of squares in an ANOVA of the linear model (i.e. an r2 value).