Abstract
Stomata regulate the supply of CO2 for photosynthesis and the rate of water loss out of the leaf. The presence of stomata on both leaf surfaces, termed amphistomy, increases photosynthetic rate, is common in plants from high light habitats, and rare otherwise. In this study I use optimality models based on leaf energy budget and photosynthetic models to ask why amphistomy is common in high light habitats. I developed an R package leafoptimizer to solve for stomatal traits that optimally balance carbon gain with water loss in a given environment. The model predicts that amphistomy is common in high light because its marginal effect on carbon gain is greater than in the shade, but only if the costs of amphistomy are also lower under high light than in the shade. More generally, covariation between costs and benefits may explain why stomatal and other traits form discrete phenotypic clusters.
Introduction
Stomata are microscopic pores formed by a pair of guard cells primarily located on the leaf surface of land plants. Their density and aperture on a leaf control the CO2 supply to leaf interiors and the rate of water lost through transpiration (recently reviewed in Sack & Buckley, 2016). Higher densities and/or larger pores allow more CO2 into the leaf, increasing photosynthetic rate, but also increasing transpiration (Farquhar & Sharkey, 1982). As the balance of CO2 and water demand and supply shifts through time and space, stomata respond over minutes to daily environmental variation, throughout the life of a single plant, and over long periods of evolutionary time (Wolfe, 1971; Woodward, 1987; Royer, 2001; Beerling & Royer, 2011; Milla et al., 2013; McElwain & Steinthorsdottir, 2017).
A less appreciated aspect of stomata is that most leaves have all their stomata on the lower (usually abaxial) surface of the leaf, termed hypostomy, while some have them on both surfaces, termed amphistomy (Metcalfe & Chalk, 1950; Peat & Fitter, 1994; Muir, 2015; Drake et al., 2019). Although amphistomy is rare in general, it is common among high light plants (Salisbury, 1927; Mott et al., 1984; Peat & Fitter, 1994; Bucher et al., 2017; Jordan et al., 2014; Muir, 2018; Drake et al., 2019). Why is amphistomy common in high light habitats but rare elsewhere? Amphistomy creates a second parallel pathway for CO2 diffusion into the leaf, which should increase photosynthesis especially when there is a lot of resistance to diffusion in the mesophyll (Parkhurst, 1978; Gutschick, 1984; Jones, 1985; Parkhurst & Mott, 1990). We might then expect amphistomy to be common, but it is not, implying some cost of amphistomy. Amphistomy also increases transpiration by forming a second boundary layer conductance for water transport (Foster & Smith, 1986, this study), but it is not clear if this tradeoff, or some other, explains variation in stomatal ratio. To evaluate these hypotheses and generate testable predictions, we need theory to predict how trait optima change across environments, both plastically and adaptively. These are classic evolutionary questions.
Stomata are also a fascinating and useful system for understanding phenotypic evolution. Land plants, like all major groups, can thrive in vastly different niches because of their diverse forms and functions, adaptations that evolved over millions of years. Less appreciated, but equally important in the study of phenotypic evolution, is that organisms occupy a small fraction of the feasible phenotypic space that could evolve in principle. This is true of stomata, as I will explain below. Why do some trait values rarely or never evolve? Three broad hypotheses explain why certain phenotypes can be rare or even absent from nature: 1) Developmental inaccessibility - a trait value is physically possible and would be favored by selection, but cannot evolve because the developmental system prevents the right genetic variation from arising; 2) Rare environments - a trait value is physically possible and would be favored by selection, but is rare because the environment that favors it is itself rare; and 3) Selection - a trait value is physically possible but is universally less fit than other trait values. Often, these hypotheses might be referred to as different phenotypic constraints (Arnold, 1992), but this terminology can be fraught with confusion and competing interpretations. In this paper, I focus on evaluating hypothesis 3, but address others throughout. Developmental inaccessibility could be important if mutations that initiate stomatal development on the upper leaf surface cause it to have the same stomatal density and size as the lower surface. This would make it easy to evolve amphistomy from hypostomy, but difficult to evolve different stomatal densities on each surface. It is also not hard to imagine that if there are discrete niches in the environment, then trait values should cluster around values best suited to those niches (Fig. 1D-F). It is more difficult to explain why trait values would cluster when the underlying environment is continuous because this implies that intermediate phenotypes are not favored in intermediate environments (Fig. 1G-I). This pattern would imply a nonlinear relationship between trait optima and environmental gradients.
The ratio of stomatal densities on the upper surface to the sum of both surfaces (hereafter termed “stomatal ratio”), is a great system for studying why traits cluster because the distribution of this trait is highly clustered and we have mathematical tools to predict the optimal trait value in different environments. Stomatal ratio forms three main trait clusters in angiosperms (Muir, 2015): hypostomy (stomatal ratio = 0); complete amphistomy (stomatal ratio = 0.5); and hyperstomy (aka epistomy, stomatal ratio = 1). There are relatively few species with intermediate values, though they do exist and there is genetic variation, suggesting that development does not preclude the evolution of intermediate trait value (Muir et al., 2014a,b, 2015). Few plants (mostly aquatic) are hyperstomatous (epistomatous), so I focus on the “bimodal” pattern describing two clusters, hypo- and amphistomy. Intermediate environments that favor intermediate stomatal ratios might be rare (Fig. 1D-F) or there may be a threshold-like relationship between the environment the trait optimum (Fig. 1H). To evaluate these hypotheses requires predictions about the relationship between the environment and trait optima.
Optimality models provide an independent way to predict the relationship between environments and trait optima against which we can compare observations of the natural world. They are an important part of identifying adaptive variation because”concordance between [optimality] model[s] and nature suggests adaptation” (Olson & Arroyo-Santos, 2015). Optimality models have a long history in successfully explaining plant form and function (Givnish, 1986, 1987), especially with stomata (Cowan & Farquhar, 1977; Buckley et al., 2017b). Optimality models based on physics and chemistry are combined with a “goal” function to generate testable predictions about how traits should vary if organisms are adapted to their environment. If optimality models predict phenotypes that do not exist in nature, this might suggest developmental inaccessibility or rare environments prevent the phenotype from evolving. Optimality models may also fail if the “goal” function, assuming it is anything other than fitness, is misspecified.
In this study, I use optimality models to predict stomatal conductance and stomatal ratio across light gradients to evaluate under what conditions, if any, we would expect phenotypic clustering to evolve along a continuous environmental gradient (hypothesis 3). I also evaluate models on their ability to predict other, independent empirical observations. Ideally, a single model should account for all of the following observations: 1) amphistomy is rare (Metcalfe & Chalk, 1950; Peat & Fitter, 1994; Muir, 2015; Drake et al., 2019); 2) amphistomy is more common in high light environments (Salisbury, 1927; Mott et al., 1984; Mott & Michaelson, 1991; Peat & Fitter, 1994; Jordan et al., 2014; Bucher et al., 2017; Muir, 2018; Drake et al., 2019); 3) amphistomy is associated with higher stomatal density (Beerling & Kelly, 1996; Muir, 2018), which is often a proxy for operational stomatal conductance (Franks & Beerling, 2009); and 4) stomatal ratio is bimodal (see above). Amphistomy is also more common in herbs than woody plants (Muir (2015, 2018), but see Drake et al. (2019)), but I do not address this observation here.
I examine three models with increasing complexity (Models 1–3). Model 1 assumes no extrinsic “cost” of amphistomy. It asks simply whether a tradeoff between carbon gain and water loss can explain the aforementioned empirical observations. Model 2 adds an extrinsic, ad hoc cost of amphistomy but is agnostic about the mechanism underlying this cost (see Discussion). Finally, Model 3 assumes that the extrinsic cost of amphistomy is not constant, but covaries with light gradients.
Materials and Methods
I used biophysical and biochemical models of leaf temperature and photosynthesis to solve for the optimal stomatal conductance and stomatal ratio across different environments. The details of the leaf temperature and photosynthetic models are described elsewhere (Muir, 2019a,b), so I briefly summarize their structure here. A glossary of model inputs and outputs can be found in Tables 1 and 2, respectively. Values of photosynthetic temperature response functions and fixed parameters are described in Tables S1 and S2 – S4, respectively.
Leaf temperature model
I modeled equilibrium leaf temperature using energy budget models (recently reviewed in Gutschick, 2016) implemented using the R package tealeaves version 1.0.1 (Muir, 2019b). Given a set of leaf parameters, environmental parameters, and physical constants, leaf energy budget models find the leaf temperature (Tleaf) such that the net energy flux in W m−2 is balanced: where Rabs is the absorbed radiation, Sr is infrared re-radiation, H is sensible heat loss, and L is latent heat loss. Absorbed radiation and infrared re-radiation are largely determined by the environment and not affected by stomatal traits. Leaf traits, especially leaf size, strongly impact sensible heat flux (H), but stomatal traits do not affect these properties directly. Stomatal traits strongly affect the total conductance to water vapor (gtw), which is proportional to the latent heat lost (L) as liquid water vaporizes and exits the leaf as a gas: tealeaves models the latent heat of vaporization (hvap) as a linear function of temperature (Muir, 2019b; Nobel, 2009). dwv is the water vapor pressure differential from the inside to the outside of leaf in units of mol m−3:
I assume the leaf interior is fully saturated (pleaf is the saturation water vapor pressure at Tleaf), where the saturation water vapor pressure is a function of temperature calculated using the Goff-Gratch equation (Vömdel, 2016). The vapor pressure of air is the product of the relative humidity (RH) and pair, the saturation water vapor pressure Tair.
gtw is the sum of the parallel lower (usually abaxial) and upper (usually adaxial) conductances in units of m s−1, which is the convention in leaf energy budget models:
The conductance to water vapor on each surface (indexed as j) is a function of parallel stomatal (gsw,j) and cuticular (guw,j) conductances in series with the boundary layer conductance (gbw,j). The stomatal and cuticular conductances on the lower surface are:
Note that the total leaf stomatal and cuticular conductances (gsw and guw respectively) are in units of µmol H2O m−2 s−1 Pa−1 in keeping with conventions of photosynthetic models (see below). In the above equations, these values are converted to units of m s−1 using the ideal gas law for the leaf energy budget model. Stomatal conductance is partitioned among leaf surfaces depending on stomatal ratio (SR). When SR = 0, all conductance is on the lower surface; when SR = 1, all conductance is on the upper surface; when SR = 0.5, conductance is evenly divided across surfaces. Cuticular conductance is assumed equal on each leaf surface, though this is probably not true in nature (Karbulková et al., 2008). The corresponding expressions for the upper surface are:
The boundary layer conductances for each surface differ because free convection differs on each surface (Foster & Smith, 1986): d is the leaf characteristic dimension in m, a physiologically relevant measure of leaf size because it determines heat and mass transfer (Taylor, 1975; Leigh et al., 2017). Dw is the diffusion coefficient of water vapor in air as a function of temperature in units of m2 s−1:
Each surface has its own unitless Sherwood number (Sh) that is a mix of free and forced convection:
The Nusselt number (Nu) is a dimensionless number for heat transfer (Monteith & Unsworth, 2013). Free convection dominates when the Archimedes number (Ar) is greater than 10; forced convection dominates when Ar ≪ 0.1 (Nobel, 2009). Forced convection is probably most common in nature (Jones, 2014), but free convection can be important for large leaves at low wind speeds (see Muir, 2019b, for further detail). Because free convection depends on gravity, horizontally oriented leaves will exchange latent heat differently depending on how transpiration through stomata is distributed between surfaces. Dh is the diffusion coefficient of heat in air a function of temperature in units of m2 s−1, calculated following Eqn 10 with Dh,0 substituted for Dw,0 (Table 1).
Transpiration rate (mol H2O m−2 s−1) is the product of the total conductance to water vapour (Eqn 4) and the water vapor gradient (Eqn 3):
Foster & Smith (1986) previously demonstrated that amphistomatous leaves transpire more water than hypostomatous leaves at low wind speeds, holding total gsw constant. To illustrate this result, I analyzed a similar model using tealeaves for hypostomatous (SR = 0), intermediate (SR = 0.25), and amphistomatous (SR = 0.5) leaves. I varied wind speed between 0 and 2 m s−1 at two light levels, photosynthetic photon flux density (PPFD) = 500 (shade) and 1500 (sun) µmol quanta m−2 s−1. I fixed other leaf parameters as absortivity of shortwave radiation (αs) = 0.5, absorbtivity of longwave radiation (αl) = 0.97, d = 0.1 m, gsw = 2 µmol H2O m−2 s−1 Pa−1, guw = 0.1 µmol H2O m−2 s−1 Pa−1. I fixed other environmental parameters where: atmospheric pressure (P) = 101.3246 kPa, relative humidity (RH) = 0.5, albedo (r) = 0.2, and air temperature (Tair) = 25 °C. Physical constants are described in Table 1. I calculated the ratio of transpiration for an intermediate or amphistomatous leaf (Ej) compared to that of hypotstomatous (Ehypo) leaf in the same environment:
Photosynthesis model
The photosynthesis package version 1.0.1 (Muir, 2019a) implements the Farquhar-von Caemmerer-Berry biochemical model of C3 photosynthesis (Farquhar et al., 1980), which has been reviewed extensively elsewhere (e.g. Sharkey et al., 2007). Following the treatment of Buckley & Diaz-Espejo (2015), the photosynthetic demand rate (AD) is the minimum of Rubisco-, RuBP regeneration-, and TPU-limited assimilation rates:
Km is the Michaelis-Menten constant:
J is a function Photosynthetic photon flux density (PPFD), obtained by solving the equation:
The photosynthetic supply rate (AS) is the product of the total conductance to CO2 (gtc [µmol CO2 m−2 s−1 Pa−1]) and CO2 drawdown (Cair − Cchl):
To facilitate modeling differentiated upper and lower leaf anatomies, photosynthesis allows users to partition boundary, cuticular, stomatal, and mesophyll conductances separately to each surface (similar to Jones, 1985). On surface j, there are two parallel conductances, the cuticular conductance (guc,j) and the in-series conductances through mesophyll (gmc,j), stomata (gsc,j), and boundary layer (gbc,j). Following rules for circuits (Nobel, 2009), the total conductance for surface j is:
To simplify the formula, I substitute resistance for conductance above. Boundary layer conductances to CO2 are calculated as described above for water vapor, but accounting for the different diffusivity of CO2 and water vapor in air (see Supporting Information for detail). The mesophyll (gmc) conductance is partitioned between layers using the following definitions: gmc is the total leaf conductance through meosphyll, partitioned to lower or upper leaf portions based on the partitioning factor kmc. The cuticular conductance to CO2 (guc) is converted from that for water vapor (guw, see Eqns 6, 8) as described in the Supporting Information.
I modeled photosynthetic temperature responses following Bernacchi et al. (2002) and Buckley & Diaz-Espejo (2015). Values of temperature-dependent parameters are provided at 25 °C as input (Table 1) and computed at Tleaf (Table 2) to determine the photosynthetic rate. The photosynthetic rate A at a given Tleaf is determined by solving for the Cchl that balances photosynthetic supply and demand rates (AD = AS).
Parkhurst (1978), Gutschick (1984), and Jones (1985) previously demonstrated that amphistomatous leaves should photosynthesize more than hypostomatous leaves holding other factors constant. To illustrate this result, I used the photosynthesis package to model photosynthetic rate for hypostomatous (SR = 0), intermediate (SR = 0.25), and amphistomatous (SR = 0.5) leaves. I varied Tleaf between 5 and 40 °C at two levels of gsw, 1 (low) and 4 (high) µmol H2O m−2 s−1 Pa−1. I fixed other leaf parameters as gmc,25 = 3 µmol CO2 m−2 s−1 Pa−1, guc = 0.1 µmol CO2 m−2 s−1 Pa−1, d = 0.1 m, Jmax,25 = 150 µmol CO2 m−2 s−1, ϕJ = 0.331, Rd,25 = 2 µmol CO2 m−2 s−1, θJ = 0.825, Vcmax,25 = 100 µmol CO2 m−2 s−1, Vtpu,25 = 200 µmol CO2 m−2 s−1. I fixed other environmental parameters where: Cair = 41 Pa, O = 21.27565 kPa, P = 101.3246 kPa, PPFD = 1500 µmol quanta m−2 s−1, RH = 0.5, Tair = Tleaf, and u = 2 m s−1. Physical constants are described in Table 1.
Optimization of stomatal traits
Biophysical and biochemical models like those implemented in tealeaves and photosynthesis help understand structure-function relationships, but cannot by themselves predict ecological and evolutionary variation. Optimality models with a defined “goal” function make testable predictions about ecological and evolutionary responses to the environment (Givnish, 1986). In plant physiology, optimality models often assume that plants will modify stomatal traits through acclimation (within generations) or adaptation (between generations) to maximize carbon gain minus costs (usually water loss) that have a carbon exchange rate (Cowan & Farquhar, 1977; Buckley et al., 2017b). Assuming a marginal water cost of carbon gain λw [mol H2O mol−1 CO2], the total carbon gain rate per area to maximize is:
This can be thought of as a profit – carbon gain minus water loss multiplied by a water-to-carbon exchange rate – to be maximized. The optimal solution will be where . The cost of water increases with the inverse of λw. For consistency in units, E in this equation must be converted from mol to µmol H2O m−2 s−1 during optimization. Traditionally, optimization models find the gsw that optimizes carbon gain and water loss, but other traits and other costs can be added for multivariate optimization. Since SR also affects carbon gain and water loss, I jointly find the optimum of both stomatal traits, denoted gsw,opt and SRopt. I also included an extrinsic cost of upper stomata (λSR [Pa−1]) in some models (see below): λSR must have Pa in the denominator so that has units µmol CO2 m−2 s−1. The cost of amphistomy is proportional to the inverse of λSR. When , this implies that stomatal conductance through the upper surface incurs some additional cost compared to the same conductance through the lower surface (see Discussion). I refer to as an ‘extrinsic’ cost of amphistomy because it is an ad hoc assumption and not an intrinsic part of the mechanistic model. Since the model does not specify mechanistically how this cost arises, I chose values of that yielded nontrivial results, but these values are arbitrary and their realism needs to be tested with experiments.
I developed an R package leafoptimizer to integrate leaf energy budget models in tealeaves and C3 photosynthesis models in photosynthesis and solve for optimal stomatal traits. leafoptimizer takes leaf parameters, environmental parameters, carbon costs, and physical constants as input (Table 1). leafoptimizer uses the R package optimx (Nash & Varadhan, 2011; Nash, 2014) to numerically solve for the trait optima by iteratively finding 1) the equilibrium Tleaf then 2) the E, A, and net carbon balance (Eq. 29) at that Tleaf until net carbon balance is maximized. For larger leaves under high light and warm temperatures, gsw,opt was often unrealistically high to cool leaves down closer to the optimum for photosynthesis (results not shown). Therefore, I set the maximum gsw,opt to 16.43 µmol H2O m−2 s−1 Pa−1, equal to gsc = 10 µmol CO2 m−2 s−1 Pa−1). Following Sharkey et al. (2007), I use units for conductance that do not change with with atmospheric pressure because they include Pa in the denominator. Often, conductance is reported in units of mol m−2 s−1 in the physiological literature. When atmospheric pressure is 100 kPa (which is approximately true near sea level), the nominal conductance in pressure-independent units (µmol m−2 s−1 Pa−1) is 10× greater than the value in units of mol m−2 s−1.
A current version of leafoptimizer is available on GitHub (https://github.com/cdmuir/leafoptimizer). The version used for this manuscript (0.0.1) is archived on Zenodo (https://zenodo.org/). I will continue developing the package and depositing revised source code on GitHub between stable release versions. Other scientists can contribute code to improve leafoptimizer or modify the source code on their own installations for a more fully customized implementation. A future publication will more fully describe the package and its potential applications. leafoptimizer depends on several other R packages: checkmate (Lang, 2017), crayon (Csárdi, 2017), dplyr (Wickham et al., 2018), glue (Hester, 2018), furrr (Vaughan & Dancho, 2018), future (Bengtsson, 2018), ggplot (Wickham, 2016), magrittr (Bache & Wickham, 2014), plyr (Wickham, 2011), purrr (Henry & Wickham, 2018a), rlang (Henry & Wickham, 2018b), stringr (Wickham, 2018), tibble (Müller & Wickham, 2019), tidyr (Wickham & Henry, 2018), tidyselect (Henry & Wickham, 2018c), and units (Pebesma et al., 2016).
Model 1: no extrinsic cost of amphistomy
Amphistomy increases E most at low wind speed and in large leaves (Foster & Smith, 1986, this study), conditions most common in forest understories where amphistomy is rare (Salisbury, 1927; Peat & Fitter, 1994; Muir, 2018). Amphistomy also increases A more under high light when CO2 limits photosynthesis (Jones, 1985; Mott et al., 1984). Therefore, I hypothesized that the increased cost of E and decreased photosynthetic benefit could drive the empirical observation that amphistomy is more common in high light environments (Salisbury, 1927; Mott et al., 1984; Mott & Michaelson, 1991; Peat & Fitter, 1994; Jordan et al., 2014; Bucher et al., 2017; Muir, 2018; Drake et al., 2019). To test whether this hypothesis is plausible, I solved for gsw,opt and SRopt across a light gradient (PPFD = 100 – 2000) at low (0.2 m s−1) and moderate (2 m s−1) wind speeds for small (d = 0.004 m), medium (d = 0.04 m), and large (d = 0.4 m) leaves. These values were chosen to ensure that free convection would be important at low wind speeds (see Results). The cost of water was mol CO2 mol−1 H2O. This value is close to that estimated for forbs and grasses under well-watered conditions (0.000981, Manzoni et al. (2011)), which is appropriate here because these functional types vary more in stomatal ratio than woody plants (Muir, 2015, 2018) and this study does not evaluate the effects of drought stress, which would increase . The extrinsic cost of upper stomata was 0. Other model variables and parameters are described in Table S2. Biochemical parameters at 25 ° C for the photosynthesis model roughly match the average and range of values from global plant surveys (Rogers et al., 2017).
Model 2: extrinsic cost of amphistomy
A fitness cost of upper stomata would explain the rarity of amphistomy in nature (Metcalfe & Chalk, 1979; Peat & Fitter, 1994; Muir, 2015, 2018; Drake et al., 2019). Model 1 tests whether a cost emerges instrinsically as a result of how stomatal ratio affect A and E. In this model, I add an exstrinsic cost to upper stomata by varying Pa. Higher (lower λSR) corresponds with a higher cost of conductance through upper stomata. Other parameters were the same or similar to Model 1 (Table S3). Because low versus high biochemical parameters Jmax,25 and Vxmax,25 had little qualitative effect (see Results), I used a single intermediate value for Models 2 and 3.
Model 3: extrinsic cost of amphistomy covaries with light
Covariation between fitness costs and benefits can generate threshold-like clines because there is a very narrow window of environments in which intermediate phenotypes are optimal. I tested this by covarying PPFD and , otherwise using the same parameter values as in Model 2 (Table S4). PPFD varied between 73 – 1927. I selected values that weakly, moderately, or strongly covaried with PPFD. varied the least (0.667 – 1.333) under the weak-covariance scenario and the most (0.002 – 1.998) under the strong-covariance scenario. In all cases, I used bivariate Gaussian covariance stucture, but adjusted the range of , as depicted in Fig. S2.
Source code for these simulations is available on GitHub (https://github.com/cdmuir/stomata-light) and archived on Zenodo (https://zenodo.org/).
Results
Amphistomy increases transpiration and CO2 assimilation
Output from tealeaves and photosynthesis packages recapitulate previous work demonstrating that amphistomy increases transpiration (E, Fig. 2A) and photosynthetic CO2 assimilation (A, Fig. 2B). When free convection is important at low wind speed and/or large leaf size, amphistomatous leaves have up to 1.5 times greater E than a hypostomatous leaf in the same conditions. The difference in E between stomatal ratio phenotypes is less when forced convection prevails at higher wind speeds. Amphistomatous leaves increase photosynthetic rate, all else being equal, by providing an additional parallel pathway for CO2 diffusion. Interestingly, leaves with intermediate phenotypes (stomatal ratio [SR] = 0.25) increase photosynthetic rate nearly as much as completely amphistomatous leaves (SR = 0.5, Fig. 2B).
Model 1: Amphistomy is almost always favored when there is no cost of upper stomata
In this model, I used leafoptimizer to solve for the gsw,opt and SRopt that optimally balances A and E across a range of environmental conditions (Table S2), given a cost of water, but no extrinsic cost of upper stomata.
In almost all areas of parameter space, the additional A associated with amphistomy outweighs the increased E (Fig. 2). A greater fraction of stomata on the lower surface can be beneficial only when reduced transpiration heats the leaf up closer to the optimum for photosynthesis (Tleaf ≈ 25°C) given the temperature response parameters assumed in this study [Fig. 2B, Table S1]). This only occurred at suboptimal air temperatures for large leaves in still air at low to moderate irradiance (Fig. 3). Forced convection dominated heat and mass transfer in smaller leaves or leaves in moving air (Figs. 3, S1). Only with the transition to free convection in large leaves and still air does reducing the conductance on the upper surface dramatically decrease transpiration (Fig. 2A). However, this beneficial effect of having lower stomatal conductance on the upper surface goes away under high irradiance because Tleaf rises toward the optimal temperature for photosynthesis. Hence, amphistomy is always favored at high irradiance when there is no extrinsic cost of upper stomata (Fig. 3). Biochemical parameters had little qualitative effect on the results (Fig. S3).
Model 2: an extrinsic cost of amphistomy produces correlations with light
Model 1 demonstrated that without an extrinsic cost, amphistomy is nearly always optimal. However, under the same leaf and environmental parameters as Model 1, an extrinsic cost leads to many situations in which hypostomy or intermediate SR are optimal (Fig. 4A). Under low light, hypostomy is better unless the cost of amphistomy is very low, but under high light, SRopt depends strongly on . When the cost is low, an intermediate SRopt occurs at most light levels; when the cost is high, SRopt is always near 0 (hypostomy). This model also predicts some covariation between SRopt and gsw,opt. At low light, both values are predicted to be low; at high light, both values are higher (Fig. 4).
Model 3: low costs of amphistomy at high light can produce threshold-like clines
Compared to Model 2, covariation between costs of amphistomy and light produced stronger threshold-like clines between light and SRopt (Fig. 5). With strong covariance, complete hypostomy (SRopt = 0) was optimal under low light and high ; complete amphistomy (SRopt = 0.5) was optimal under high light and low . The correlation between SRopt and gsw,opt was similar to Model 2.
Discussion
I used three optimality models based on the biophysics and biochemistry of leaf temperature and photosynthesis to predict stomatal ratio (SRopt) and conductance (gsw,opt) across light gradients. I draw three substantial conclusions about the evolution of stomatal traits that inform more general questions about phenotypic evolution.
First, a tradeoff between increased photosynthetic CO2 assimilation (A, 2B) and water loss (E, Fig. 2A) does not explain why amphistomy is rare because the benefits almost always outweigh the costs (Model 1, Table 3). Previous modeling and experiments already demonstrated the physiological effects of amphistomy on A and E (Parkhurst, 1978; Gutschick, 1984; Foster & Smith, 1986; Parkhurst & Mott, 1990; S̆antrůc̆ek et al., 2019), but these insights have not been combined for optimality modeling. Hypostomy is sometimes optimal at low wind speed, low/partial sun, and suboptimal temperatures (Fig. 3, S1) because decreased E brings Tleaf closer to its optimum. However, these restrictive conditions are probably not common in nature; even light wind speeds greater than 1 m s−1 would completely eliminate this effect (Fig. 3).
Second, an extrinsic cost of amphistomy produces a cline between light and SRopt (Model 2, Fig. 4). Under the same parameters in Model 1, no such cline is predicted. A previous phenomenological model also suggested that the cost of amphistomy is important (Muir, 2015), but could not distinguish between an “intrinsic” (Model 1) and “extrinsic” (Models 2 and 3) cost. The leaf temperature and photosynthesis models in this study indicate that the tradeoff between A and E is not the mechanism explaining stomatal ratio, but future mechanistic models of other processes effected by stomatal ratio (e.g. hydraulic conductance outside the xylem (Buckley et al., 2015, 2017a; Drake et al., 2019)) may reveal an ‘intrinsic’ cost. Model 2 also explains why stomatal ratio and conductance positively covary along light gradients (Muir, 2018). Both SRopt and gsw,opt are beneficial under high light because the marginal benefit of increased CO2 supply is greater under high light. I am assuming here that stomatal density is a proxy for operational stomatal conductance (Franks & Beerling, 2009). Generally, stomatal density increases with light up to an intermediate value then decreases slightly (Poorter et al., 2019), consistent with model predictions here (Figs. 4B, 5B). However, in real plants, many other traits change in response to light which are forced to remain constant in the model, so this correspondance between model predictions and experiments may be spurious. Overall, this model indicates that optimizing both density and distribution of stomata on a leaf may help plants fully take advantage of high light and should be considered together in future analyses of light responses.
Third, only when the cost of amphistomy covaries with light does a threshold-like trait-environment relationship emerge (Model 3, Fig. 5). Model 2 explains other empirical observations (Table 3) but fails to explain why intermediate stomatal ratio trait values are rare in nature. Under that model, intermediate values should be common. Only by coupling a benefit of increased A under high light with a low cost of amphistomy in the same environment do we predict discrete clusters of hypo- and amphistomatous leaves (i.e. bimodality). Covariation between costs of amphistomy and light may be the only way in this modeling framework to get phenotypic clusters when the underlying environmental gradient is continuous. I used light as an environmental gradient based on a priori hypotheses, but covariation between the cost of amphistomy and another environment or trait could produce qualitatively similar results. For example, amphistomy increases A more in leaves with high resistance to mesophyll CO2 diffusion (Parkhurst, 1978). Covariation between and that trait could also produce a similar effect, but would not necessarily explain why amphistomy is common in high light environments.
The goals of optimality models are to accomodate existing observations and generate new testable predictions. Model 3 accomodates existing observations, but is complex and therefore important to evaluate with future empirical tests of its predictions. In particular, the model implicates the importance of covariation between costs and benefits of amphistomy. Hypostomy is favored in low light with low costs of amphistomy, but high light only favors amphistomy (SRopt = 0.5) when costs are also low. This is important because some proposed costs probably do not covary with light gradients this way, while others likely do. For example, amphistomy can dehydrate the palisade mesophyll when there is strong evaporative demand (Buckley et al., 2017a; Drake et al., 2019), but this cost should be stronger, not weaker, under high light. Furthermore, when leaves can optimize both SR and gsw simultaneously, amphistomatous leaves have lower gsw,opt and hence lower evaporative demand than hypostomatous plants holding all else constant (Fig. 4). Amphistomy may also be costly if it increases susceptibility to foliar pathogens that are more likely to land on the upper surface of a horizontally oriented leaf (Gutschick, 1984; McKown et al., 2014). Because many pathogens need a wet leaf microclimate to germinate and grow, a leaf in high light that dries faster is less likely to experience this cost than one in the shade. Hence, if pathogens are the primary cost of amphistomy, then this cost should be higher in shady habitats and lower in sunny habitats, consistent with the assumptions of Model 3. Future work should focus on identifying the abiotic and biotic cost(s) of upper stomata at different light levels under natural conditions. We also need to evaluate how often the distribution of light values is unimodal in nature (hypothesis 2) and the role of developmental constraints on stomatal evolution (hypothesis 1).
There are several important limitations of this study that will need to be addressed in future work. Currently leafoptimizer only optimizes stomatal traits while other traits are held constant. But traits such as leaf size, mesophyll conductance, Jmax/Vcmax acclimate and evolve too. If all these traits could vary together in the model, different patterns might emerge. For example, high light favors thick leaves to capture more photons and greater investment in photosynthetic biochemistry, traits that make increased CO2 supply more advantageous. In this case, a greater benefit rather than increased cost might explain why amphistomy is common at high light. Furthermore, this study did not exhaustively explore relevant parameter space. It is possible that further exploration may reveal patterns not identified here. For example, I only used a single set of temperature response functions, even those these vary within and between species (Medlyn et al., 2002; Slot & Winter, 2017). However, this limitation does not qualitatively change the result that amphistomy only significantly affects evaporation, and hence leaf temperature, when leaf size is large, wind speed is almost zero, and there is relatively high sunlight. These conditions are not common in nature. Different temperature response parameters that change optimum leaf temperature would alter the range of air temperatures in which hypostomy would helps keep leaf temperature closer its optimum under restrictive parameter spece. The model also uses bulk leaf properties of temperature and photosynthesis at one time point, ignoring spatial variation within the leaf and temporal variation in the environment, which might yield different predictions (Buckley et al., 2017a; Earles et al., 2019). Finally, carbon gain and water loss are not fitness, which is what natural selection cares about. Future theoretical and empirical studies should integrate plant survivorship and reproduction with stomatal function.
Amphistomy is rare despite the fact that it increases photosynthetic rate. Why? Optimality models show this is not because the increased carbon gain is offset by additional water loss. Instead, an additional cost of amphistomy, yet to be identified, must explain why it is rare. Optimality models also predict that amphistomy is common in high light habitats not just because it increases carbon gain but also because the costs of amphistomy are lower. Covariation between costs and benefits may also explain why stomatal ratio forms discrete phenotypic clusters.
Funding
This work was supported by startup funds from the University of Hawai’i.
Supporting Information
Photosynthetic temperature responses
I calcualted gmc, Γ*, Jmax, KC, KO, Rd, Vcmax, and Vtpu at Tleaf (Table 2) based on an assumed value at 25 °C (Table 1) and temperature response paramters from (Bernacchi et al., 2002, Table S1). Parameters with an exponentially increasing response to temperature were modeled as: and those with a humped-shaped response were modeled as:
Ea and Ed are the enthalpies of activation and deactivation, respectively, and Ds is the entropy. Tref is a reference temperature (25 °C) in K; Tleaf is a reference temperature in °C.
Parameter conversions in leafoptimizer
Because of their differing origins and uses, leaf energy budget and photosynthesis models sometimes employ different units for the same parameter. As standalone packages, tealeaves and photosynthesis honor these conventions, but leafoptimizer must convert between them. Here I document these conversions.
As noted in the Materials and Methods section, conductance values are converted from m s−1 (tealeaves) to µmol m−2 s−1 Pa−1 (photosynthesis) using the ideal gas law:
Conductance to water vapor and CO2 are interconverted using the the gc2gw() and gw2qc() functions:
Incident shortwave radiation (Ssw [W m−2], tealeaves) is interconverted with PPFD [µmol quanta m−2 s−1] (photosynthesis) following Gutschick (2016) using the functions sun2ppfd() and ppfd2sun(). Shortwave radiation is (at first approximation) the sum of photosynthetically active radiation (PAR) and near-infrared radiation (NIR):
Most sources (e.g. Jones, 2014) assume that SPAR = SNIR for sunlight, so fPAR = 0.5. To convert PAR to PPFD, divide by the energy per mol quanta. assuming Eq = 220 kJ mol−1 quanta for PAR: tealeaves uses stomatal ratio (SR), while photosynthesis uses a partitioning factor ksc. These are automatically interconverted as:
Acknowledgements
I would like to thank Joseph Stinziano and an anonymous reviewer for valuable feedback on this work.
Footnotes
I fixed an error in units of the cost of amphistomy and improved the clarity of the text.