^{ 1 }School of Biological Sciences, University of Utah, Salt Lake City, UT 84112 USA

^{ 1 }

^{ * }Date of publication: 22 Dec 2018

Stomatal response to environmental variation has a major influence on ecosystem carbon, water, and energy fluxes, and thus is important for global carbon and water cycles (Berry ^{st} century (Field

Stomatal response to drought – some combination of increasing atmospheric water demand through vapor pressure deficit (VPD) or decreasing soil water availability through falling soil water potential (_{s}) – is a key component of plant drought responses (Tuzet

The vast majority of ecosystem models use empirical algorithms of stomatal conductance (Ball

Optimal stomatal theories show promise for predicting stomatal conductance in future climates based on linking physiological processes with an evolutionary optimization to maximize plant fitness in varying environments (Medlyn _{N}) minus the risk of hydraulic damage driven by low plant water potential (Θ(

This CM optimization (also called the “gain-risk” algorithm in Sperry ^{2} of 0.11-0.25) over the classic marginal water use efficiency optimization when tested against a dataset of 34 woody plant species spanning global biomes (Anderegg

Several key unresolved questions remain, however, concerning the CM optimization at longer time-scales and in varying environments. In particular, do these optimal stomatal parameters vary either 1) directly in response to drought drivers (i.e. VPD and/or _{s}) and/or 2) in lagged response to drought drivers? This allows the examination of an interesting question: are stomata Bayesian in that they update their hydraulic risk “shadow price” (in the CM optimization, this shadow price is the marginal increase in carbon cost with a marginal drop in water potential – the partial derivative of the cost function; dΘ(

I used a subset of five species from a recently published dataset that compiled concurrent measurements of leaf-level gas exchange and plant water potentials (Anderegg _{s}) and the key environmental variables needed to drive the plant model: photosynthetically active radiation (PAR), leaf-to-air vapor pressure deficit (VPD), carbon dioxide concentration at the leaf surface (Ca), predawn and midday leaf water potentials. In addition, the critical plant traits of the stem hydraulic vulnerability curve and the maximum carboxylation capacity (V_{cmax}) of leaves is known for these species and presented either in the original study (e.g. Limousin

To quantify the daily and seasonal shadow price (dΘ(*et al.* (2018) that couples hydraulic transport of water from the soil to the atmosphere with photosynthesis via the CM optimization. The model uses four equations that describe the dynamics of water transport and photosynthesis and a fifth equation from the CM optimization that allows solving for stomatal conductance at a given time-point with a given set of environmental conditions (for full details of the model, see Anderegg _{c }(CO_{2}/rubisco limitation) and _{j} (light limitation):

(1)

where, C_{i} is the internal leaf CO_{2} concentration, Γ^{*} is the CO_{2} compensation point, K_{c} and K_{o} are Michaelis-Menten coefficients of the carboxylation and oxidation reactions performed by rubisco, O_{i} is the internal partial pressure of oxygen, J is the potential maximum rate of electron transport, calculated as in Medlyn _{d} is the rate of dark respiration calculated using a Q10 functional form. I used the standard implementation of the photosynthetic model presented in the freely available R package “plantecophys” .

For the second equation, the model uses a simplified version of Fick’s Law:

(2)

where g_{s} is stomatal conductance of the leaf to water vapor (mol m^{-2} s^{-1}), 1.6 accounts for the difference in diffusion coefficients between water vapor and CO_{2}, and C_{a} is the partial pressure of CO_{2} in the atmosphere. These equations assume that cuticular conductance is negligible and boundary layer and mesophyll conductances are much larger than stomatal conductance, which is likely reasonable for these species and environmental conditions (see Anderegg

The third and fourth equations describe the water transport through the hydraulic continuum from soil to leaf and the water lost through stomatal conductance (transpiration):

(3)

where E is transpiration, e_{a} is the vapor pressure of water in the atmosphere at ambient temperature and relative humidity, and e_{s} is the vapor pressure of the saturated air space inside the leaf. Steady-state E is found by integrating the conductance function K() from soil water potential (here, measured plant pre-dawn water potential) to the leaf water potential (Sperry

(4)

where _{s} and _{L}

The fifth and final equation in the model is the optimality equation that allows solving the system of equations to find a predicted stomatal conductance:

(5)

where (

Θ/

g_{s}) was fit as the following function of leaf water potential:

(6)

where *et al.* (2016) and Anderegg *et al.* (2018), the

I used a Markov Chain Monte Carlo (MCMC) process to find the shadow price (parameter _{s} and environmental drivers at a given time-point), an initial g_{s} of 0.010 mol m^{-2} s^{-1 }was guessed. At that guess of g_{s}, the photosynthetic rate was then solved for using Equations 1-2 and the leaf water potential was solved for using Equations 3-4. The initial g_{s} was incremented slightly (+0.001 mol m^{-2} s^{-1}) and the new A_{N }and _{L }were found, allowing the calculation of the right hand side of Equation 5. Equation 6 is then solved with the current guesses of parameters _{s}.

The MCMC was implemented to minimize the SSE between predicted and observed g_{s}. After preliminary testing to find the ideal step size (typically 0.1-1 depending on species) in guesses of parameters

To examine diurnal changes in the shadow price, I performed the MCMC on each day for the three species with adequate daily data (>8 measurements within a day). I extracted the maximum likelihood (lowest SSE) parameter _{s }if plant water potential has equilibrated with the soil (e.g. minimal nighttime transpiration) (Donovan

For

To determine which variables best predicted daily shadow price variation, I performed a model selection procedure with two sets of variables: 1) current-day variables only and 2) current and lagged variables together. Because model selection can be adversely affected by strongly collinear predictor variables, I first removed collinear predictor variables using a standard procedure (Anderegg

To assess if the shadow price varied between wet and dry periods (a seasonal analysis), I performed the MCMC to estimate parameter

Daily estimates of the slope of the cost function (_{s} (p>0.05 for all regressions; Fig. 1, Fig. 2) in the three species with adequate data to estimate daily parameters. There was substantial variation in the slope across days and larger variation in the range of slopes across species, but this variation was unrelated to increasing water stress. The marginal cost increased slightly but insignificantly at higher VPDs in all species and increased slightly but insignificantly as soil water potential declined in two of three species (Fig. 1, Fig. 2).

The slope of the stomatal cost function is uncorrelated with vapor pressure deficit (VPD; kPa) across the tree species

The slope of the stomatal cost function is uncorrelated with vapor pressure deficit (VPD; kPa) across the tree species

The slope of the stomatal cost function is uncorrelated with predawn leaf water potential (Ψ_{pd}; MPa_{) across the tree species Pinus edulis (a), Juniperus monosperma (b), and Quercus douglasii (c). Each point is a day and the color of points is the vapor pressure deficit (color bar) for that day and redder colors indicate higher VPD values. Lines are the OLS best fit and are not statistically significant.}

The slope of the stomatal cost function is uncorrelated with predawn leaf water potential (Ψ_{pd}; MPa_{) across the tree species Pinus edulis (a), Juniperus monosperma (b), and Quercus douglasii (c). Each point is a day and the color of points is the vapor pressure deficit (color bar) for that day and redder colors indicate higher VPD values. Lines are the OLS best fit and are not statistically significant.}

While the predictive ability of the slope of the cost function was fairly low overall in both species with a multitude of daily data (

titre du tableau
^{2}
Current
LWPhigh
0.17
0.04
N measurements
Date
Current+lag
LWPhigh
0.25
0.03
N measurements
Date
LagLWPlow
Current
LWPhigh
0.12
0.1
Date
Current+lag
LWPhigh
0.23
0.01
Date
LagVPDmax

Model selection results for predicting the shadow price (parameter

Examining seasonal variation in cost functions, I observed moderate seasonal differences in cost functions in the two gymnosperm species (p<0.01), but not in the three angiosperm species (Fig. 3). Uncertainty in cost functions was somewhat larger in the wetter periods than drier periods, especially in the tropical angiosperm species (Fig. 3). A sensitivity analysis with an alternate vulnerability curve for *et al.*, 2017) showed predicted gs values that were very similar to the ones presented here (R^{2}=0.99). One potential reason for the detection of seasonal differences in the gymnosperm species may be due to biome-level differences in climate, whereby the gymnosperm species experienced a much larger variation of water potential and much greater declines in water potential during the drier periods (Fig. 4).

Despite slight seasonal differences in the gymnosperm species, the predictive ability of a single set of CM parameters at a species level was quite similar to the model allowing for seasonal differences in CM parameters (R^{2}
_{species}=0.48; R^{2}
_{seas}=0.50; Fig. 5). Indeed, the predictive differences between the models was minimal; the root mean squared error difference to the models was 0.002 mol m^{-2} sec^{-1 }(RMSE_{species}=0.077; RMSE_{seas}=0.075). Predictive ability was strongest in the two gymnosperm and the temperate oak species and substantially lower in the two tropical species.

Seasonal estimates of the stomatal cost function slope parameter in the wet (blue) and dry (red) periods for

Seasonal estimates of the stomatal cost function slope parameter in the wet (blue) and dry (red) periods for

Range in predawn leaf water potential for all species in the wet (blue) and dry (red) periods for

Range in predawn leaf water potential for all species in the wet (blue) and dry (red) periods for

Observed versus predicted stomatal conductance (gs; mol m^{-2} sec^{-1}) for all five species where a separate set of parameters are used for wet and dry periods for each species (left) or a single set of parameters is used for each species (right). Colors show the density of points with gray and blue as low density and red to yellow as highest density. Red line is the ordinary least squares regression best fit and black line is the 1:1 line.

Observed versus predicted stomatal conductance (gs; mol m^{-2} sec^{-1}) for all five species where a separate set of parameters are used for wet and dry periods for each species (left) or a single set of parameters is used for each species (right). Colors show the density of points with gray and blue as low density and red to yellow as highest density. Red line is the ordinary least squares regression best fit and black line is the 1:1 line.

Predicting stomatal conductance across multiple time-scales is a central aim of many vegetation models and may be informed by optimal stomatal models. I analyzed the parameters of an optimal stomatal model at multiple time-scales and reached three central conclusions. First, the key parameter of the CM optimal stomatal model – the marginal cost of water potential – does not change predictably in response to VPD and _{s} in the species analyzed here. Second, there is some evidence for the influence of lagged variables on stomatal model parameters, indicating stomata might be behaving in a somewhat Bayesian manner, though the evidence is not strong. Finally, while slight seasonal changes in parameters were detected in two of five species, a single set of species-level parameters worked nearly as well for prediction of stomatal conductance across all time-scales.

The longest and most detailed datasets of the two conifer species revealed that including previous date’s lowest leaf water potential (

In the CM optimization, the cost that stomata aim to avoid is formulated as an instantaneous carbon cost, but most likely includes multiple aspects of hydraulic “risk” that could play out over longer time periods (Wolf

The CM optimization shows substantial promise for inclusion into next-generation ecosystem models because plant hydraulic transport provides a mechanistic way to simulate drought stress on plants from easily measurable traits and because the CM optimization showed the highest predictive ability of stomatal conductance in previous analyses (Anderegg et al. 2018). The findings here that seasonal differences in parameters are minimal and do not greatly affect accuracy of prediction of leaf-level stomatal conductance lend further support. Intra-specific variation and plasticity in plant drought responses and hydraulic transport are likely to be important in many contexts and may not be captured in the relatively short (1-2 year) datasets included here. Plasticity or variation can be incorporated hydraulic-enabled ecosystem models through hydraulic trait variation, which will allow the CM optimization to flexibly predict stomatal conductance in a wide array of environments.

I thank Y.S. Lin and B. Medlyn for compiling the original database from which several species’ stomatal conductance data were obtained for this study and J. Limousin, B. Choat, and D. Baldocchi for sharing data. I further thank S. MacAdam and N. Martin for their insightful reviews. W.R.L.A. acknowledges funding from the University of Utah Global Change and Sustainability Center, NSF Grant 1714972, and the USDA National Institute of Food and Agriculture, Agricultural and Food Research Initiative Competitive Programme, Ecosystem Services and Agro-ecosystem Management, grant no. 2018-67019-27850.