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Unexpected Connections between Humidity and Ion Transport Discovered using a Model to Bridge Guard Cell-to-Leaf Scales

Stomatal movements depend on the transport and metabolism of osmotic solutes that drive reversible changes in guard cell volume and turgor. These processes are defined by a deep knowledge of the identities of the key transporters and of their biophysical and regulatory properties, and have been modelled successfully with quantitative kinetic detail at the cellular level. Transpiration of the leaf and canopy, by contrast, is described by quasi-linear, empirical relations for the inputs of atmospheric humidity, CO 2 , and light, but without connection to guard cell mechanics. Until now, no framework has been available to bridge this gap and provide an understanding of their connections. Here we introduce OnGuard2, a quantitative systems platform that utilizes the molecular mechanics of ion transport, metabolism and signalling of the guard cell to define the water relations and transpiration of the leaf. We show that OnGuard2 faithfully reproduces the kinetics of stomatal conductance in Arabidopsis, its dependence on vapor pressure difference ( VPD) and on water feed to the leaf. OnGuard2 also predicted with VPD unexpected alterations in K + channel activities, and changes in stomatal conductance of the slac1 Cl - channel and ost2 H + -ATPase mutants which we verified experimentally. OnGuard2 thus bridges the micro-macro divide, offering a powerful tool with which to explore the links between guard cell homeostasis, stomatal dynamics and foliar transpiration. we cells 2012; Wang al., 2012). We show that this next-generation platform, OnGuard2, faithfully reproduces stomatal dependence on VPD and predicts emergent characteristics, including elevations in [Ca 2+ ] i , unexpected alterations in the K + channel activities, and altered VPD responses in the slac1 Cl - channel and ost2 H + -ATPase mutants of Arabidopsis. We validate each of these predictions experimentally. The findings demonstrate that OnGuard2 provides a reliable representation of the mechanistic link between guard cell membrane transport and foliar transpiration.


INTRODUCTION
Stomata provide the main pathway for CO2 entry for photosynthesis and for transpirational water loss across the leaf epidermis. Pairs of guard cells surround each stoma, regulating the aperture to balance the often conflicting demands for CO2 and for water conservation. Guard cells open and close the pore, driven by osmotic solute uptake and loss-notably of K + and Cl --and by the synthesis and metabolism of organic solutes, especially sucrose (Suc) and malate (Mal) (Willmer and Fricker, 1996;Kim et al., 2010;Roelfsema and Hedrich, 2010;Lawson and Blatt, 2014; Jezek and Blatt, 2017). A number of well-defined signals, including light, CO2, and the water-stress hormone abscisic acid (ABA), modulate transport and solute accumulation to alter cell volume, turgor and stomatal aperture. Much research at the cellular level has focused on these inputs and their connection to stomatal movements, especially stomatal closing. Studies have highlighted both Ca 2+independent and Ca 2+ -dependent signalling, including elevated free cytosolic Ca 2+ concentration ([Ca 2+ ]i), cytosolic pH (pHi), protein kinases and phosphatases, that inactivate inward-rectifying K + channels and activate Clchannels and outwardrectifying K + channels to bias the membrane for solute loss (Blatt et al., 1990;Lemtiri-Chlieh and MacRobbie, 1994;Grabov andBlatt, 1998, 1999;Marten et al., 2007;Assmann and Jegla, 2016;Jezek and Blatt, 2017).
At the tissue and whole-plant levels, by contrast, attention has been drawn to inputs closely tied to photosynthesis, including transpirational water loss (E) driven by the vapor pressure difference (VPD) between the atmosphere and the intercellular space of the leaf. Stomata respond to a drop in atmospheric humidity with a reduction in aperture (Lange et al., 1971;Buckley and Mott, 2002;Shope et al., 2008). This aperture change helps preserve water, albeit often at the expense of reduced carbon fixation (Lawson and Blatt, 2014). The immediate effect of reducing atmospheric humidity-that is, raising the VPD-is to increase the rate of transpiration driven by the difference in the vapor pressure of water between the inside and outside the leaf (Δw = wleaf -wair). The internal partial vapor pressure of water is almost always inferred and assumed to remain close to saturation, wsat (Willmer and Fricker, 1996;Farquhar et al., 2001). For these reasons, stomatal conductance (g s = E/Δw) is often plotted against external relative humidity (RH = wair/wsat).
Despite the extensive body of phenomenological data, an understanding of the mechanism connecting stomatal transpiration with guard cell membrane transport is lacking. There is a clear need for a mechanistic framework that bridges the micro-macro gap between the guard cell and whole-plant transpiration. Here we introduce such a framework, building on the original OnGuard platform that encompasses guard cell transport, signalling and homeostasis and successfully models guard cells across species, including those of Arabidopsis (Chen et al., 2012;Hills et al., 2012;. We show that this next-generation platform, OnGuard2, faithfully reproduces stomatal dependence on VPD and predicts emergent characteristics, including elevations in [Ca 2+ ]i, unexpected alterations in the K + channel activities, and altered VPD responses in the slac1 Clchannel and ost2 H + -ATPase mutants of Arabidopsis. We validate each of these predictions experimentally. The findings demonstrate that OnGuard2 provides a reliable representation of the mechanistic link between guard cell membrane transport and foliar transpiration.

Rationale for the modelling approach
The majority of mechanisms that have been proposed for the stomatal response to VPD assume that the response is caused by a change in foliar water potential or a parameter related to the rate of water vapor diffusion from the leaf.
Although transpiration is affected by external wair (Δw = wleaf -wair, often expressed as the corresponding difference in the mole fractions of water vapor), the vapor pressure of water in the leaf also depends on leaf temperature, Tleaf, which alters the equilibrium between the liquid and vapor phases of water. Leaf temperature affects other processes, however, notably photosynthesis and metabolism in the mesophyll (Smith and Dukes, 2013) and guard cells (Willmer and Fricker, 1996). Not surprisingly, most studies of foliar transpiration and stomatal response to VPD have employed changes in wair at constant or near-constant Tleaf. In the natural environment, changes in temperature most often arise with solar radiation, the associated heat driving evaporation within the leaf which effectively absorbs the thermal load and facilitates transpiration to the surrounding air (Pieruschka et al., 2010). Thus, it is to be expected that, at a given air temperature, Tleaf will stabilize with near-constant irradiation, provided that water supply to the leaf is not limiting. As a first approximation, therefore, Tleaf is commonly assumed to be constant.
Beyond the drivers for evapotranspiration, most mechanistic models that have been proposed start from the premise either (i) that the guard cells respond to a chemical signal produced by evaporating site(s) distant from the guard cell (Buckley et al., 2003), or (ii) that the guard cells are supplied by liquid flow through the epidermis and evaporation occurs directly from the guard cells (Farquhar, 1978;Maier-Maercker, 1983;Dewar, 1995;Buckley, 2005). The difficulty with the first model is that no obvious signal has been identified, beyond water in the vapor phase itself, that can account for key aspects of stomatal behavior, for example the differential response of amphistomatous leaves to different VPDs on either side of the same leaf (Mott, 2007). The second model posits evaporation from the guard cells directly, leading to their water loss and stomatal closure. However, a growing body of evidence indicates that the bulk of the evaporation within the leaf occurs well away from the guard cells, dominated instead by evaporation from the vascular tissue, nearby mesophyll and pavement cells (Pieruschka et al., 2010;Rockwell et al., 2014;Buckley et al., 2017). These studies imply that water in the guard cell is maintained primarily by equilibration with water in the vapor phase within the adjacent substomatal cavity and the stomatal pore rather than by liquid flow through the cell wall and neighboring cells.
Regardless of their starting premise, virtually all models for transpiration to date have considered stomata to operate entirely on the basis of passive water flux driven by evaporation and diffusion without reference to solute transport in the guard cells and its consequences for osmotic water flux, turgor, and stomatal aperture. One exception is the hydromechanical model of Buckley, et al. (2003), which proposed a simple hyperbolic relation between the ATP concentration of the guard cells and their osmotic content. Even this model failed to incorporate guard cell solute transport explicitly, however, so obviating the intrinsic regulatory processes that are known to determine much of stomatal function (Jezek and Blatt, 2017). Furthermore, all mechanistic models to date have sought analytic solutions for endpoint or stationary states only. They therefore fail to address the wealth of information available relating to the temporal kinetics for stomatal movements and transpiration.
By contrast with analytical approaches, numerical computation allows flexibility in experimentally guided model design, and it is amenable to representing the temporal kinetics of interacting processes (Lew and Bookchin, 1986;Mauritz et al., 2009). It avoids solutions that may appear to provide simplified or explicit equations for a process but that sterilize the predictive power of model implementation. We previously used iterative numerical computation in developing the OnGuard platform to model membrane transport, signalling, and homeostasis in guard cells (Chen et al., 2012;Hills et al., 2012). OnGuard uses small increments in time to introduce deviations away from the previous steady state, calculating and logging the dynamic adjustments of ion flux, compartmental composition, membrane voltage and guard cell volume that result with each step in time. It uses the sets of constants, non-linear differential equations, and parameters contained within the model to calculate ion flux, while obeying the fundamental physical constraints of mass and charge conservation. It generates a new total solute content for the guard cell at the end of each time interval, using this value to calculate the new total and compartmental cell volumes and ion concentrations, the guard cell turgor, and stomatal aperture. Thus, like other computational approaches (Lew et al., 1979;Lew and Bookchin, 1986;Mauritz et al., 2009), OnGuard does not transit between two pre-defined states but, instead, begins from a starting, or reference, state (Chen et al., 2012;Hills et al., 2012), from which a new state evolves over each time increment.
Models resolved with this platform successfully recapitulated a wide range of known stomatal behaviors, including transport, diurnal changes in aperture, their dependencies on extracellular pH, KCl, and CaCl2 concentrations (Chen et al., 2012), and oscillations in [Ca 2+ ]i and membrane voltage thought to facilitate stomatal closure (Blatt, 2000;McAinsh and Pittman, 2009;Minguet-Parramona et al., 2016).
OnGuard models demonstrated true predictive power, for example in resolving the mechanisms behind the counterintuitive alterations in K + channel activity uncovered in the slac1 Clchannel mutant of Arabidopsis which we subsequently validated experimentally . Finally, OnGuard models provided a mechanistic understanding of [Ca 2+ ]i oscillation frequency and its relation to solute flux during stomatal closure (Minguet-Parramona et al., 2016).
OnGuard incorporates all the fundamental properties of the transporters at the plasma membrane and tonoplast that determine the osmotic relations of the guard cell, the salient features of Suc and Mal metabolism, and the essential H + and Ca 2+ buffering characteristics that have been described in the literature (Hills et al., 2012; Jezek and Blatt, 2017). The OnGuard platform accommodates spectrally-discrete irradiation, and it utilizes photosynthesis to energize membrane transport and Suc and Mal metabolism (Chen et al., 2012;Vialet-Chabrand et al., 2017). Within OnGuard, two key assumptions only are relevant to modelling evapotranspiration [we direct the reader to the detailed descriptions of Hills, et al. (2012) andChen, et al. (2012)]. These assumptions are that the water permeabilities of the plasma membrane and tonoplast are high, so water flux normally equilibrates rapidly with respect to ion transport, and that the apoplast of the cell wall space surrounding the guard cells provides a near-infinite reservoir for solutes uninfluenced by material entering or emanating from the guard cell. The first assumption is founded on reliable data (Hill and Findlay, 1981;Willmer and Fricker, 1996;Maurel, 1997;Chaumont and Tyerman, 2014;Maurel et al., 2015); it allows guard cell volume and turgor to be resolved from the Van't Hoff relation with stomatal aperture from empirical relations of steady-state experimental measurements (Hills et al., 2012).
The second assumption greatly simplifies a treatment of ion transport across the plasma membrane, leaving its ion composition and the associated osmotic potential, Ciso, to be defined by the user. Although the platform does not account explicitly for ion exchange with the surrounding pavement cells, the latter assumption is consistent with the presumed role of the pavement cells in buffering apoplastic solutes transported by the guard cells and with the known and substantial cation buffer capacity of the cell wall matrix (Bowling, 1987;Bush and McColl, 1987;Marschner, 1995;Willmer and Fricker, 1996;Buckley et al., 2003).
These assumptions offer a starting point from which to accommodate stomatal responses to VPD, allowing us to develop the next-generation OnGuard2 platform from the core of knowledge subsumed within the original software. To bridge the gap from the microscopic physiology of solute transport in the guard cell to the macroscopic relations of foliar and whole-plant water flux, therefore, three key issues only need be addressed, namely formalisms (1) that couple the guard cell to transpiration, (2) that account for water delivery to the leaf, and (3) that address water flux across the guard cell membrane. Below we summarize the reasoning and solutions behind each in turn. All the novel model developments are detailed in Appendix 1 at the end of this article together with Appendix Table 1 that summarizes the abbreviations for the various parameters and constants.

Connecting the vapor phase with guard cell water
To accommodate evidence of evaporation away from the guard cells (Pieruschka et al., 2010;Rockwell et al., 2014;Buckley et al., 2017), we assume that water in the guard cell is maintained by equilibration with water in the vapor phase within the adjacent substomatal cavity and the stomatal pore. No system undergoing steady-state flux can be fully at equilibrium, but it is an adequate approximation for evaporation, water vapor diffusion, and exchange inside a leaf. Like Peak and Mott (2011), we assume a constant leaf temperature, Tleaf, but provide both Tleaf and the air temperature, Tair, as parameters for run-time manipulation by the user, and we define a site, p, internal to the leaf and near the stomatal pore, with which guard cell water equilibrates (Figure 1). The advantage of this description is that it couples guard cell response to VPD through the vapor pressure of water at p, wp. We stress that the p site concept does not imply a single, physical site with which guard cell water exchanges but represents a hypothetical position at a constant fraction of the resistance for water vapor diffusion to the outside (Peak and Mott, 2011). It implies a gradient in the vapor pressure of water between the sites of evaporation within the leaf and the outside, with the gradient extending through the stomatal pore and into the substomatal cavity. In other words, p defines a position along a standing gradient in water vapor that extends into the leaf air space.
In situ, much of the free (unoccluded) guard cell surface is exposed to the air within and immediately adjacent to the pore area between the guard cells. It follows that vapor exchange with the water in the guard cell will be subject to this gradient.
We place p to the inside of the stomatal pore and interpret it to reflect the average of this vapor pressure gradient over the guard cell with which it exchanges. Similarly, we define this exchange to take place with the potential of liquid water averaged over the guard cell surface. Beyond exchange with the guard cell, the p concept does not specify the tissues from which the bulk of evaporation occurs and accommodates evaporation from the mesophyll as well as from the inner surface of the epidermis (Buckley et al., 2017). Since the conductivity for liquid water to the stomatal pore will be much lower than that for water vapor within the leaf air space, cells at different points in the leaf along the transpiration path will exchange with different partial pressures of water in the vapor phase.
It is important to note that Peak and Mott (2011) define the vapor phase equilibration of guard cell water without reference to the guard cell wall or its osmotic content. Thus, their original formulation cannot accommodate membrane transport between the apoplast and cytosol or water flux across the plasma membrane.
Clearly, vapor exchange with liquid water of the guard cell must first occur between water in the guard cell wall and the air in the stomatal pore and substomatal cavity.
We identify the external wall space as a compartment with a finite osmotic load, consistent with the vast literature of the past half-century on cell walls and the ionic relations of plant cells [cf. (Findlay and Hope, 1976;Taiz, 1984;Bush and McColl, 1987;Grignon and Sentenac, 1991;Sanders et al., 2002;Holdaway-Clarke and Hepler, 2003;Jezek and Blatt, 2017)]. Thus, OnGuard2 incorporates the vapor phase at point p and its thermal equilibrium with the liquid phase water surrounding the guard cell so that the volume of the guard cell VT = QT/( P/RT + Ciso) where Ciso = Ciso o + 1/νL . ln(wsat/wp) .
[ those of adding osmotic solute to the cell wall. Furthermore, the very large value of 1/νL (=56000 mmol/l) multiplies the natural logarithm of the ratio wsat/wp. So, even small increases in this ratio with a decline in wp will have substantial effects on Ciso, thereby affecting guard cell volume, turgor pressure, and stomatal aperture. As wp defines the water vapor pressure within the substomatal cavity adjacent to the guard cells, it also determines the rate of water loss from the leaf. From Fick's Law, the rate of water vapor diffusion through the stomatal pore is proportional to the driving force across the pore, Δws = wp -wair. A consideration of the diffusion pathway suggests that wp normally will be close to wleaf (≈wsat), which is often used as a convenient approximation to calculate stomatal conductance. However, depressing wp when water delivery to the leaf is reduced will affect the rate of transpiration proportionally, all other factors being equal.
We resolve wp assuming that water vapor lost via diffusion through the stomatal pore will be replaced by evaporation of water delivered to the leaf and diffusion to p ( Figure 1). Thus, in the steady-state, wp can be sought through the balance between diffusion through the pore and delivery from the sites of evaporation (see Appendix 1). As noted above, evaporation within the leaf is a complex function of factors that include water delivery through the xylem and its transfer across the evaporative surfaces of the surrounding mesophyll and near-by epidermal cells (Sack and Holbrook, 2006;Rockwell et al., 2014;Buckley et al., 2017). The evaporative surface area, AW (Figure 1), is thought to vary substantially with hydraulic conductance through the xylem (Rockwell et al., 2014) as well as environmental factors, including wair (Buckley et al., 2017). In OnGuard2, this factor becomes a user-defined parameter, the relative water feed (RWF), that encapsulates water delivery to the leaf, its conductance and evaporative distribution within the leaf, and hence any water stress arising through its restriction.

Water flux across the guard cell plasma membrane
In operation, OnGuard2 calculates and logs the dynamics of changes in flux, compartmental composition, membrane voltage and all other variables using sets of nonlinear differential equations while obeying the fundamental physical constraints of mass and charge conservation. With each small step in time, the routine resolves a new pseudo-steady-state that satisfies these physical constraints in relation to flux.
The approach is common to modelling strategies that use iterative numerical calculation to integrate the temporal kinetics of a complex network of processes, but it generally imposes a practical constraint on model construction: By establishing a new steady-state within each time interval, it presents a barrier to any temporal dynamics that might be encoded separate to the iteration cycle.
In the original OnGuard platform, we resolved guard cell volume, turgor and stomatal aperture from the change in QT using the Van't Hoff relation assuming that the water permeabilities of the plasma membrane and tonoplast are high. In other words, water permeability across the guard cell membrane was assumed not to limit VT. There are situations, however, in which water flux is slowed (Shope et al., 2008;Prado et al., 2013;Chaumont and Tyerman, 2014). Indeed, there is good evidence to suggest that the bulk of water flux is mediated by aquaporins at the plasma membrane and their activity is vital for stomatal movements in the intact leaf (Yang et al., 2006a;Yang et al., 2006b;Grondin et al., 2015). Thus, a treatment of stomatal transpiration must also provide for water flux and its regulation across the guard cell plasma membrane.
Within any series of iterative time increments, it is possible to generate a secondary relaxation over multiple increments by interleaving a time-dependent alteration to the incremental change in the process output without imposing internal complexity to the computational cycle. To a first approximation, water flux will follow a change in osmotic potential across the membrane with a simple exponential relaxation to a new steady state (Murai-Hatano and Kuwagata, 2007;Chaumont and Tyerman, 2014;Maurel et al., 2015). Therefore, to enable independent control of water flux in OnGuard2 consistent with a reduced osmotic permeability, we assigned water flux to a population of aquaporins with an exponential pseudo-rate constant, f, to adjust the increment in cell volume with each iteration (see Appendix 1). We add the adjusted volume increment to the previous cell volume and then calculate the new cytosolic and vacuolar volumes, solute concentrations and turgor pressure.
Thus, f serves as a factor, related to membrane water permeability, that defines the relaxation in water flux relative to water equilibration with ion transport. In so doing, water flux remains an output of modelling.
Finally, we implemented ligand sensitivity to f (see Appendix 1), much as previously described for ion transport (

OnGuard2 predicts a steep dependence of gs at limiting hydraulic feed
OnGuard2 predicted changes in stomatal aperture and g s that faithfully reproduced experimental measurements across a wide range of conditions in water feed and VPD ( Figure 2). The model also demonstrated a dependence on temperature (Supplemental Figure 1) that is broadly consistent with the curvilinear relationships previously reported for stomatal aperture across a number of species (Wilson, 1948;Stalfelt, 1962;Spence et al., 1984;Willmer and Fricker, 1996;Urban et al., 2017).
Step increases in VPD (decreasing %RH outside) reduced steady-state turgor, aperture and g s with a concurrent increase in total guard cell osmolarity; these values reversed when VPD was returned to its starting value. With RWF set to 100-equivalent to a well-watered plant-simulations showed a shallow dependence on VPD in aperture and g s , even when the vapor content of the air outside the leaf was reduced to 10% RH. When RWF of the leaf was reduced, notably below 40, the sensitivity to VPD was strongly enhanced (

Counterintuitive alterations in K + channel activities
In simulation, increasing VPD enhanced the outward-rectifying (IK,out) and reduced the inward-rectifying (IK,in) K + channel currents (Figure 4a . As predicted, we found (Figure 4c,d) that a 300 mOsM rise outside led to a rapid and prolonged reduction in IK,in and an increase in IK,out. Washing out the osmotica led to a recovery in IK,out. Significantly, washout was also accompanied by a large overshoot in IK,in that remained elevated for 5-10 min before declining towards the initial current characteristics.
We analysed the steady-state K + currents using a Boltzmann equation of the form where gmax is the maximum ensemble conductance, V is the voltage, V1/2 is the voltage at which g=0.5 . gmax, δ is the voltage sensitivity coefficient of the channel, EK is the equilibrium voltage for K + , and F is the Faraday constant. Analysis of IK,in showed that adding and removing osmolyte evoked shifts in V1/2 of -12±4 mV and +9±3 mV, and altered gmax by -40±3 % and +73±5 %, respectively, relative to the initial values (Figure 4e), as expected for [Ca 2+ ]i ranging over roughly 200 and 400 nM (Grabov and Blatt, 1999;. The same manipulations affected IK,out, but in a manner consistent with the reversible changes in pHi and [K + ]i with cell volume (Figure 4f). The timescales of these recordings, typically 20-30 min, precluded measurements to full recovery. However, the reversible increase in IK,out with osmotic strength and the counterintuitive enhancement in IK,in on its washout validate the model predictions. We also tested the rise in [K + ]i on adding 300 mOsM solute using tail currents of the K + channels to quantify the change in the equilibriium voltage for K + (Blatt and Clint, 1989). The analysis indicated a small, but significant displacement of -4±1 mV (n=24), indicating roughly a 30 mM increase in [K + ]i with the osmotic challenge and again consistent with the modelling.

OnGuard2 predicts altered VPD responses in the slac1 and ost2 mutants
The ost2  To test these predictions, we carried out gas exchange measurements on leaves, recorded K + currents under voltage clamp, and measured [Ca 2+ ]i and pHi in vivo following injections of the Ca 2+ -and H + -sensitive dyes Fura2 and BCECF, respectively (Grabov andBlatt, 1998, 1999;. As predicted, the ost2 mutant showed greatly slowed stomatal kinetics in response to step increases in VPD and its recovery (Figure 6a however, together with the analysis of the ost2 mutant, they demonstrate the predictive power of OnGuard2 and its ability to bridge the micro-macro gap in stomatal transpiration and its response to VPD.

DISCUSSION
Stomatal aperture and conductance are affected by the vapor pressure difference between inside and outside the leaf, especially when water delivery to the leaf is limiting (Lange et al., 1971;Willmer and Fricker, 1996;Pieruschka et al., 2010). Although widely recognized, previous efforts failed to explain the mechanistic connection to the microscopic processes of guard cell ion and water transport.
Recent studies support the idea that water in the guard cell wall is normally maintained by, and equilibrates with, the vapor phase within the intercellular air We stress that our findings do not rule out additional mechanisms, such as might occur through more subtle changes in phosphorylation (Grondin et al., 2015) or ABA synthesis (McAdam and Brodribb, 2016 Like our previous work with OnGuard , the predictions derived from OnGuard2 and their experimental validations underscore an intimate connection of the K + channels with the mechanistically separate activities of the SLAC1 Clchannel and AHA1 H + -ATPase. Most important, however, our present study demonstrates the capacity of a single, computational framework to connect the microscopic processes of membrane transport in the guard cell with the macroscopic water relations of the leaf and whole plant. This development paves the way to addressing many other aspects of stomatal physiology, and it is certain to yield a greater understanding of fundamental and mutual connections between plant water relations and guard cell membrane transport.

Growth and whole-plant physiology
Arabidopsis thaliana Col-0 wild-type, slac1-1, ProSLAC1:SLAC1complemented slac1-1 mutant, designated pSLAC1, and ost2-2 mutant plants were grown and gas exchange measurements were carried out using LiCOR 6400XT and 6800 gas exchange systems (Lincoln, USA) as described previously (Merlot et al., 2007; with plants preadapted to dark. Relative water content of the soil was determined using a HH2-ML3 moisture sensor (Delta-T Devices, Cambridge UK). All plants were analysed on at least three separate days at the same time of the relative diurnal cycle and were normalised for leaf area using ImageJ v.1.51 (rsbweb.nih.gov/ij/). Unless otherwise noted, all measurements were carried out at 25ºC.

Gene expression analysis
Total RNA was extracted from mature leaves and transcript levels determined by quantitative PCR as before  using unique primers (Supplemental Table 1

Guard cell electrophysiology, [Ca 2+ ]i and pHi recording
Currents from intact guard cells were recorded under voltage clamp using double-and triple-barrelled microelectrodes and Henry's EP suite (Y-Science, Glasgow, UK) during continuous superfusion with 10 mM KCl in 5 mM Ca 2+ -MES buffer, pH 6.1 ([Ca 2+ ] = 1 mM) as before . Voltage was typically clamped in cycles with a holding voltage of -100 mV and steps either to voltages from -120 to -240 mV (IK,in)   . Fluorescence was corrected for background before loading, dye injections were judged successful by visual checks for cytosolic dye distribution and stabilisation of the fluorescence signals, and signals were calibrated as before (Grabov and Blatt, 1999;.

OnGuard2 modelling
OnGuard2, was constructed to introduce explicitly water feed to, and evapotranspiration from, the leaf, leaf and air temperatures, as well as water flux across the guard cell plasma membrane as described in Appendix 1. The construction used the core of the original OnGuard libraries for solute transport, signaling, and homeostasis (Chen et al., 2012;Hills et al., 2012; with separate assignments of blue and red light (Vialet-Chabrand et al., 2017).
OnGuard2 models for wild-type Arabidopsis and the slac1 and ost2 mutants were driven through a diurnal 12:12 h light:dark cycle as described previously (Chen et al., 2012;Blatt et al., 2014), and all model outputs were derived from this cycle. Leaf and air temperatures, Tleaf and Tair respectively, were held at 25 ºC unless otherwise noted. Relative water feed (RWF) and mid-day steps in wair were superimposed on this cycle. As in the original formulation of OnGuard, lightsensitivity was assigned solely to primary, energy-dependent transport and to sucrose synthesis (Chen et al., 2012;Hills et al., 2012;. To simulate the two mutants, in the corresponding models we removed the transporter component of the SLAC1 current for slac1, and we removed the [Ca 2+ ]i sensitivity of the plasma membrane H + -ATPase for ost2 Blatt et al., 2014). All other parameters were fixed as in the wild-type model. Arabidopsis are available for free download from www.psrg.org.uk.

Statistics
Results are reported as means ±SE of n observations with significance determined by Analysis of Variance (ANOVA), as appropriate, with post-hoc analysis (Student-Neumann-Keuls and Tukey), and are indicated at P<0.05 unless otherwise stated. Note that models built on ordinary differential equations, such as those of OnGuard2, will faithfully reproduce a given set of outputs time and again for any one set of model parameters. Statistical analysis of these outputs is therefore meaningless.

Accession Numbers
Sequence data from this article can be obtained from the Arabidopsis Genome Initiative databases under the following accession numbers:  Supplemental Appendix 1. OnGuard2 Arabidopsis model parameters.

General construction and operation of the OnGuard2 platform
OnGuard2 include global parameters that specify the geometry of the stoma and its spatial relations to the underlying structure of the leaf-notably the fractional air space within the leaf and its dimensions-as well as water feed to the leaf, and the partial pressure of water vapor in the air outside the leaf. We give the user access to leaf and external air temperatures, previously fixed to a standard 25ºC (298ºK) and used in all calculations involving temperature. For completeness, we also introduce temperature dependence for Suc and Mal metabolism with a user-defined parameter for Q10 (Segel, 1993).
OnGuard2 uses the same iterative computational strategy that was applied successfully in our original implementation of the OnGuard platform (Chen et al., 2012;Hills et al., 2012;Blatt et al., 2014) and, in the past, in other physiological systems (Lew et al., 1979;Lew and Bookchin, 1986;Mauritz et al., 2009). In operation, OnGuard2 calculates and logs the dynamic adjustments of ion flux, compartmental composition, and membrane voltage using the sets of constants, non-linear differential equations, and parameters contained within the model, while obeying the fundamental physical constraints of mass and charge conservation. Outputs of the ion concentrations of each compartment, plasma membrane and tonoplast voltages, as well as stomatal aperture, g s , and related variables are available in real time onscreen. The user is able to log these outputs, along with the flux through each transporter and metabolic reaction, in comma-delimited format that can be read by spreadsheet programs such as Microsoft Excel and SigmaPlot.
Finally, the user has access to real-time displays of the current-voltage relations for each of the charge-carrying transporters at the plasma membrane and tonoplast, as well as the total membrane current, all of which are also available for download.
In implementing OnGuard2, we bridge the gap from the guard cell to wholeplant evapotranspiration with three critical developments encapsulated in mathematical formulations (1) that couple the guard cell to transpiration, (2) that account for water delivery to the leaf, and (3) that address water flux across the guard cell plasma membrane. The reasoning behind each development and the associated assumptions are set out in the main text. Here we summarize the mathematical formulations that are incorporated within the new OnGuard2 libraries of constants, non-linear differential equations and parameters. A complete list of abbreviations for the various parameters and constants will be found at the end of this description in Appendix Table 1.

Connecting the vapor phase with guard cell water
We define stomatal dynamics initially by the gain and loss of osmotically active solutes and water within the guard cell via transport across the plasma membrane and tonoplast, and by the synthesis and breakdown of Suc and Mal; solute content is then related through experimentally-defined equations to guard cell volume, turgor, and stomatal aperture (Hills et al., 2012). The tonoplast cannot sustain hydrostatic pressure differences, and its permeability to water is high (Hill and Findlay, 1981;Willmer and Fricker, 1996;Maurel, 1997;Nobel, 2012;Chaumont and Tyerman, 2014;Maurel et al., 2015), so we assume that the osmotic pressure difference between the vacuole and cytoplasm is always zero.
In the original OnGuard platform, the turgor pressure at each instant in time was assumed to arise from the osmotic pressure of the guard cell so that where VT is the volume of the guard cell with the osmotic load QT under the prevailing pressure constraints of the cell wall and the surrounding epidermal or subsidiary cells, and R and T have their usual meanings (Appendix Table 1 below).
In this formulation, Ciso, defined the sum of the concentrations of all osmotically active solutes in the extracellular space, the apoplast, and the water potential of the apoplast, was assumed to be determined entirely by these solutes. Solving for VT, VT = QT/(P/RT + Ciso) . [A2] Evaporation from the leaf commonly has been envisaged to be driven by the water vapor gradient between the liquid-water interface within the leaf and the air and solving for VT gives VT = QT/(P/RT + 1/νL . ln(wsat/wp) + Ciso o ) . [A5] Comparing

Water flux across the guard cell plasma membrane
In implementing Equation [A2] in OnGuard, we assumed that water permeability across the guard cell plasma membrane normally does not limit VT and that osmotic and turgor pressures balance within each iteration cycle. Water flux was therefore defined by where ΔVT is the change in guard cell volume calculated over the time interval Δt.
There are situations, however, in which water flux is restricted, such as may be associated with a reduced osmotic permeability through the regulation of plasma membrane aquaporins (Yang et al., 2006b;Chaumont and Tyerman, 2014;Grondin et al., 2015;Maurel et al., 2015). In these situations, the change in volume with each time interval will be proportional to the driving force generated during the interval. It can be shown that where Fw r is the restricted water flux, ΔVT r is the consequent change in volume over the interval Δt, and f is a non-negative, pseudo-rate constant with units of s -1 .

Equation [
A8] defines a first-order process and is consistent with trans-plasma membrane water flux that commonly approximates a simple exponential relaxation (Murai-Hatano and Kuwagata, 2007;Chaumont and Tyerman, 2014;Maurel et al., 2015).
To restrict water flux in OnGuard2, we interleave this relaxation with each iteration using ΔVT r to calculate a new value of VT n for iteration n so that VT n = VT n-1 + ΔVT r . [A9] Thus, f adjusts the change in total volume calculated from the volume of Equation [A5] to moderate the water flux and the corresponding volume change relative to VT in each iteration. We use this new value, VT n , to calculate the variables for cytosolic and vacuolar volumes, solute concentrations and turgor pressure. Finally, we implement ligand sensitivity to f as previously described for ion transport (Hills et al., 2012) so that f is scaled with one or more sets of Hill parameters assigned by the user to represent the dependence of Fw r on ligands affecting plasma membrane water flux.
Implicit to VT n in Equation [A9] is a residual osmotic potential across the plasma membrane. This residual adds to the osmotic potential generated by solute flux for balance in the next time interval. Thus, its iterative generation and carry-over accommodates an independent but slower relaxation in water flux. Clearly, when f is large, ΔVT r = ΔVT and VT n does not deviate from the value obtained with Equation

Scaling and delivering water to the leaf and water vapor to p
To scale models from the single stoma to the whole leaf, and to determine wp and g s , we assign each stoma an equal fraction of the underlying total intercellular air space within the leaf for gas exchange. The intercellular volume, VS, assigned to each stoma where the values, de and VF are, respectively, the depth of the mesophyll cell layer and the volume fraction that is air space within the leaf, and DS is the stomatal density over the leaf surface. VS is used to scale the quantity of water transpired by the whole leaf. Diffusion through the stomatal pore is defined by Fick's Law such that JW = ΔwS . DW/dS and ΔwS = wp-wair [A11a,b] where JW is the flux per unit area through the pore, dS is the depth of the stomatal pore, and DW is the diffusion coefficient for water vapor in air. The micro-macro link is achieved by scaling this flux per stoma to the whole leaf surface E S = JW . AS , and E = E S . DS [A12a,b] where E S is the transpiration through the single stoma and AS is the area of the stomatal pore. As noted previously, g s is commonly derived from E and Δw assuming This scalar therefore becomes a user-defined parameter, the relative water feed (RWF), that encapsulates water delivery to, and its conductance within, the leaf and, hence, any water stress arising through its restriction. In effect, for Aw>>AS, wp≈wsat and for Aw<<AS, wp≈wair.   and an expanded schematic near the pore is shown in (B). Liquid water from the xylem (grey cells, blue surface) feeds the surfaces of the surrounding mesophyll (green surfaces) and nearby pavement cells (dark green surfaces). The effective evaporative surface is indicated by the red dotted line (=A w ). Evaporation and vapor flux from this surface (blue arrows) maintains a high partial vapor pressure of water within the leaf and close to saturation (w sat ) adjacent this surface. As water vapor is lost through the stomatal pore (A S indicated, defined by the width and breadth of the pore), driven by diffusion towards a lower w air , a steady-state gradient is established in the partial vapor pressure of water (background shaded blue color), reducing its value at w p and at points towards the outside of the leaf. Mathematically, p represents the average of the water vapor gradient centered over the guard cell which exchanges with liquid water in the guard cell wall, thereby affecting C iso . Figure 2. OnGuard2 reproduces stomatal conductance (g s ) and its dependence on the partial vapor pressure of water in the air (w air ) and relative water feed (RWF) available for transpiration from the plant. (a) Model outputs for g s with vapor pressure difference (VPD) steps generated with steps from 85 %RH to 40 %RH, and with RWF values of 100, 20 and 10, as indicated. Model outputs for aperture, guard cell volume, turgor and osmolarity (bottom two graphs) correspond to g s with RWF=10. (b) g s from wild-type Arabidopsis with steps from 85 %RH to 40 %RH (n≥3 independent experiments in each case). Transients during the first 10 min omitted for clarity. Damped oscillations at higher relative water content (RWC) of the soil are probably associated with the Ivanoff effect and subsequent changes in turgor of the surrounding epidermal cells (Lawson and Blatt, 2014). (c) Steady-state dependence of g s on w air and RWF determined from the OnGuard2 model [surface plot, includes data in (a)] overlaid with experimental measurements (open symbols) from Arabidopsis, including the data in (b), plotting RWC as a percentage of saturation (=100). Note that a direct relationship between RWF and RWC is not implied, although water feed to the leaf will depend on water available in the soil. Figure 5. The OnGuard2 model for the ost2 mutation predicts slowed kinetics in g s and aperture with step changes in vapor pressure difference (VPD), and reduced alterations in K + channel activities with equivalent step changes in external osmolarity, compared to the wild type. (a) OnGuard2 outputs in g s and stomatal aperture for the ost2 and wild-type models with external %RH as indicated. (b,c) OnGuard2 outputs of the ost2 model for the outward-rectifying (I K,out , b) and inwardrectifying K + current (I K,in , c) before, 10 min into a step to 40 %RH (= +300 mOsM), and 5 min after its recovery (red, washout), as in Figure 4a,b. Dotted curves included from the wild-type controls in Figure 4a,b for reference. Note the substantially enhanced I K,out and reduced I K,in in ost2 compared to the wild type, predicted as a consequence of the elevated [Ca 2+ ] i and pH i . (d) OnGuard2 outputs for cytosolic-free [Ca 2+ ] i ([Ca 2+ ] i ) and pH (pH i ) before and during VPD (osmotic) steps from the wild-type and ost2 models. Note the elevated pH i and [Ca 2+ ] i of the ost2 model, the latter further enhanced on VPD (osmotic) challenge.