OnGuard, a Computational Platform for Quantitative Kinetic Modeling of Guard Cell Physiology

The Reference State Wizard

To establish a Reference State, we devised a software wizard that allows the operator to specify the underlying biophysical status of the system—standard ion concentrations in each compartment and voltages across the plasma membrane and tonoplast—and then query the model for net charge, driver ion (for plants, H⁺), and solute fluxes. The wizard allows the operator to review and edit parameters such as membrane voltage and compartment solute composition, and to determine parameters for metabolic equilibrium in Suc and Mal synthesis and catabolism; most important, the wizard gives access to pages that compare the fluxes of ionic species across each membrane and permit their balancing by adjustment to the population(s) of transporters and, as necessary, to their underlying kinetic descriptors (see Fig. 3B). Thus, the operator is able to interrogate and adjust the subsets of transporters affecting the flux of each ion to satisfy the requirements of a Reference State.

Obviously such manipulations imply a knowledge of the (likely) unit density and/or limiting current amplitudes and their kinetic parameters for each transporter against which the operator can base judgment of biological validity. For most transporters of guard cells this information is available, often to within a factor of three and in many cases with an accuracy of a few percent of the mean for the parameter (see Supplemental Tables S3–S6). Even when information for some transporters was uncertain, we found their characteristics sufficiently constrained by the properties of other processes to ensure minimal indetermination. For example, with the H⁺-ATPase as the sole pathway for H⁺ export across the plasma membrane, achieving charge and net H⁺ flux balance in the Reference State required a close match with H⁺ consumption through Mal catabolism and with the dominant H⁺ return pathways, and required similar balancing of the associated ion fluxes. Defining the H⁺-Cl⁻ and H⁺-K⁺ symporters included in our model, but for which there is limited information in guard cells (see below), required coordinating their relative contributions to this H⁺ balance. The H⁺-coupling ratios of these transporters (H⁺-Cl⁻ symport returns two H⁺ per charge while H⁺-K⁺ symport returns 0.5 H⁺ per charge; Sanders et al., 1985; Blatt and Slayman, 1987) implies a 2:1 ratio in the activities of the two currents at the free-running membrane voltage to achieve charge balance in 1:1 ratio with H⁺ export via the H⁺-ATPase. Additionally, H⁺-coupled Cl⁻ influx required an equal efflux of Cl⁻ through the sum of the anion channels, and H⁺-coupled K⁺ influx required an equal efflux of K⁺ through the outward-rectifying K⁺ channels. Finally, each of these components added a new current and therefore required balancing with opposing currents of the H⁺-ATPase and K⁺ channels such that the total membrane current is zero at the free-running voltage. The consequence was to constrain each of the currents to within a narrow range of values relative to one another and within the constraints of the known densities and current amplitudes for the H⁺-ATPase, K⁺, and Cl⁻ channels characteristic of the guard cells (see Supplemental Tables S3 and S4).

In practice, a systematic approach dictated that fluxes be balanced first at the tonoplast and then at the plasma membrane. Flux adjustment at each membrane began with nondriver ions (K⁺, Cl⁻, Ca²⁺, Mal²⁻, Suc) and culminated with fine adjustments to the H⁺-ATPases (and for the tonoplast, the H⁺-PPase) densities. Transporter numbers were assessed against the available experimental data and the process was repeated if the approximation failed. We found it sufficient to bring all of the fluxes of each of the transported species in balance to within ±10⁻¹⁵ mol s⁻¹; thereafter trial simulations achieved a stable Reference State, generally within 8 to 10 h of simulation time, and maintained this condition even when allowed to run free over periods equivalent to many months.

The Choice of Transporters and Their Parameters

Most of the relevant quantitative kinetic information available comes from studies of the guard cells of Vicia (Supplemental Tables S3–S6). Therefore, we used this cell system as our starting point for model construction. Nonetheless, a complete set of parameters for all of the pumps, carriers, and channels is not available for any one species. A comparison of transport parameters listed in the tables shows a substantial degree of similarity between species, and even between cell types. For example, the fundamental gating characteristics for the plasma membrane K⁺ channels of Vicia, tobacco, and Arabidopsis guard cells (see Supplemental Table S4) include overlapping values for gating charge (δ) and half-activation voltage (V_1/2), as well as a similar sensitivities to extracellular [K⁺], cytosolic pH, and [Ca²⁺]_i (Dreyer and Blatt, 2009). Furthermore, Supplemental Table S3 highlights the close quantitative relationships between the plasma membrane H⁺-ATPases of Vicia guard cells (Blatt, 1987a, 1988a), the giant alga Chara (Blatt et al., 1990a), and the fungus Neurospora (Gradmann et al., 1978; Sanders and Slayman, 1982; Blatt and Slayman, 1987). For these reasons we felt confident to borrow characteristics from other species—even other cell types—when information was lacking for Vicia, as necessary scaling the transporter for Vicia guard cells. For example, kinetic detail of H⁺-coupled K⁺ symport is missing for guard cells. Nonetheless, an estimate of the probable symport current can be derived from measurements of K⁺ uptake against its electrochemical gradient in Vicia guard cells (see Supplemental Table S3; Blatt and Clint, 1989; Clint and Blatt, 1989) and a current similar to that documented for the H⁺-K⁺ symport of Arabidopsis roots (Maathuis and Sanders, 1994). We could have modeled this transporter as a simple current source—that is, independent of membrane voltage—but we reasoned that the kinetic constraint of voltage was likely to be important for the dynamics of K⁺ and H⁺ balance, much as has been demonstrated in Neurospora (Blatt and Slayman, 1987). H⁺-coupled K⁺ transport was originally described in Neurospora (Rodriguez-Navarro et al., 1986; Blatt and Slayman, 1987; Blatt et al., 1987) and, although the current is typically 10-fold greater in K⁺-starved Neurospora than in the plant cells, the qualitative kinetic dependencies on external [K⁺], pH, and voltage are similar. A relative weighting of reaction-kinetic constants is available for the carrier cycle of the H⁺-coupled K⁺ symport in Neurospora (Blatt et al., 1987); hence, we used this weighting and the current amplitudes from Arabidopsis and estimated for Vicia as a guide in assigning values to the reaction-kinetic constants for the H⁺-K⁺ symport in the guard cells. Much less kinetic information is available for plasma membrane Ca²⁺-ATPases. Again, it was unrealistic to assume its voltage independence: Best estimates place the equilibrium voltage for the plasma membrane Ca²⁺ pumps near −200 mV with 1 mm [Ca²⁺] outside and, hence, within the physiological voltage range. Instead, we borrowed the carrier cycle established for the H⁺-ATPase, scaling it to estimates of the typical Ca²⁺ flux and measured values for the K_m with respect to [Ca²⁺]_i, thereby ascribing a significant dependence on membrane voltage over much of the physiological range. Finally, to avoid some uncertainties in defining parameters we concatenated transport activities associated with Cl⁻ and NO₃⁻, subsuming NO₃⁻ with Cl⁻ as a single, anionic species. Both anions contribute to the osmotic content of the guard cells, their relative presence depending to a large extent on availability, both are permeable through several of the anion channels present at the two membranes, and coupled transport for Cl⁻ and NO₃⁻ show similar thermodynamic and kinetic properties, to the extent that they are known (Sanders et al., 1989; Meharg and Blatt, 1995). By this maneuver we could draw on the available kinetic detail for H⁺-coupled NO₃⁻ transport at both membranes (Supplemental Tables S3 and S5) and avoid redundancy with less-well-defined characteristics, notably for Cl⁻ transport at the plasma membrane.

A case-by-case summary of the reasoning behind the selection of each transporter will be found in Supplemental Appendix S2. The complete set of OnGuard parameters used for the modeling described in this article will be found in Supplemental Appendix S6 and is available for download with a demonstration of the OnGuard software (see www.psrg.org.uk). Our inclusive approach taken in developing the model described below expands the number of model parameters and, correspondingly, the model complexity. It is often a choice in modeling to use a skeleton construction to determine the minimum number of components, and associated parameters, needed to simulate a particular set of biological phenomena. In this case, however, the parameter space for the ensemble of transporters is exceptionally well constrained experimentally, generally to within a factor of three and frequently with an accuracy of a few percent (see Supplemental Tables S3–S6). This information, and the fundamental requirements for charge and ionic balance, placed narrow limits on the characteristics of the remaining transporters, even when their parameterization was less certain. As we demonstrate here, and in the companion article (Chen et al., 2012), a deterministic solution is readily found that recapitulates a very wide range of physiological behaviors. Nonetheless, the OnGuard software enables the user to generate and test skeleton models with equal ease.

Closed and Open Reference States

To test the parameter space of OnGuard models and their robustness, we used the results of initial simulations as a guide in defining a pair of states of the guard cell associated with the closed and open stomata, namely the Closed and Open Reference States. The Closed Reference State was envisaged as typical of stomata in the dark, in which the guard cells retained a baseline of osmotic load and a minimum of ion flux across the tonoplast and plasma membrane. For purposes of modeling, the currents of all primary pumps (H⁺-ATPases, Ca²⁺-ATPases, H⁺-PPase) at the tonoplast and plasma membrane were assigned values of 5%, 10%, or 20% of their outputs in the Open Reference State, consistent with experimental estimates of the known light-stimulated activities of plasma membrane ATPases (Assmann et al., 1985; Shimazaki et al., 1986; Serrano et al., 1988; Goh et al., 1995, 1996; Kinoshita et al., 2001; Chen et al., 2012). Again, for the Closed Reference State we sought parameter sets such that the net fluxes of all solutes, Φ_i, across both tonoplast and plasma membrane were zero, that is ΣΦ_i = 0 for each membrane. We then used the same model parameters for the Open Reference State, allowing only the step up in primary pump currents, Suc and Mal synthesis. All other model variables were kept constant between these paired Reference States, and fine adjustments to the properties of individual transporters were introduced between simulations. Thus, Reference State pairing offered a convenient test of the capacity of the parameter ensemble to support stomatal movement across a range of common environmental variables.

As an initial test of the parameter ensemble, we explored model robustness by varying systematically the surface densities of individual transporters within bounds of ±1.3-, 2-, and 5-fold of their starting values as dictated by experimental knowledge and uncertainty. Figure 4 summarizes the results of simulations within the ±1.3-fold boundary reporting both macroscopic (aperture, osmotic content, total Ca²⁺) and microscopic (pH_i, pH_v, [Ca²⁺]_i) outputs. Displacements of each output were normalized to the corresponding value obtained with the starting parameters for purposes of comparison. In general, we found the model to be relatively insensitive to several of the predominant osmotic solute transporters at the plasma membrane. For example, a 1.3-fold increase or decrease in the population of R- (ALMT-) type anion channels (Keller et al., 1989; Meyer et al., 2010) resulted in <5% variation in any of the outputs, either in the closed or open states. Only outside the physiological limits for the channel population (see Supplemental Table S4) were variations in output returned that, in principle, might be detectable through biological experimentation (not shown). Similar sensitivities were recovered with variations in densities for the outward-rectifying K⁺ channel, and H⁺-Mal symport at the plasma membrane, and for analogs of the TPK1, TPC1, FV, VCL, and VMAL channels at the tonoplast. These results demonstrated a considerable robustness despite the corresponding ranges in densities associated with these transporters. They also suggest a substantial functional tolerance for divergent transporter densities in the guard cell.

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Figure 4.

Relative sensitivity of the OnGuard model to the component transport activities at the plasma membrane (A and B) and tonoplast (C and D) when transporter densities were varied ±1.3-fold about the starting values. Model outputs (n_i) were determined in the Closed (A and C) and Open (B and D) Reference States and normalized to the outputs with the corresponding starting values (n_o). For purposes of comparison, outputs are reported for selected macroscopic (stomatal aperture, total osmotic content, and total Ca²⁺ content) and microscopic ([H⁺]_i, [H⁺]_v, and [Ca²⁺]_i) variables. Additional details are provided in the text.

The OnGuard model proved substantially more sensitive to variations in the densities of transporters directly affecting [Ca²⁺]_i and cytosolic pH (see Fig. 4). As central signaling and energetic elements, both [Ca²⁺]_i and cytosolic pH affect an extended network of transporters at both membranes (Blatt, 2000; Schroeder et al., 2001) and, thus, were anticipated to impose greater restrictions on performance. Substantially greater relative variations in macro- and/or microscopic outputs were recovered, even with 1.3-fold changes in densities of the H⁺-ATPase, Ca²⁺-ATPase, and Ca²⁺ channel at the plasma membrane, and for the Ca²⁺-ATPase and Ca²⁺ channel at the tonoplast. Notable sensitivities, especially in cytosolic and/or vacuolar [H⁺] were evident also to changes in the densities of the H⁺-Cl⁻ and H⁺-K⁺ symporters at the plasma membrane and the CLC, NHX, and CAX antiporters at the tonoplast. Aperture and osmotic content sensitivities to H⁺-Cl⁻ symport density is consistent with its central role in anion uptake and, for reasons noted above, variations in either the H⁺-Cl⁻ symport, H⁺-K⁺ symport, or H⁺-ATPase densities had the greatest impact in the Closed Reference State, especially on pH_i balance. The roles for the endosomal cation and anion antiporters in regulating pH are broadly consistent with observation documented in the literature (Pardo et al., 2006; Padmanaban et al., 2007; Braun et al., 2010; Smith and Lippiat, 2010; Weinert et al., 2010).

For purposes of comparison, we also varied the biophysical and kinetic parameters for transporters with appreciable effects on both the macro- and microscopic parameters in Figure 4. As examples, Figure 5 summarizes the outputs from simulations with the plasma membrane Ca²⁺ channel and the tonoplast Ca²⁺-ATPase, both of which contribute to the Ca²⁺ circuit of the model. For the Ca²⁺ channel, we varied by ±2-fold each of the key gating parameters affecting its voltage sensitivity—V_1/2 and δ that define the voltage-yielding half-maximal conductance and the sensitivity of the channel gate to changes in voltage, respectively—and the apparent K_Ca and associated Hill coefficient, h_Ca, for channel inactivation by [Ca²⁺]_i. For the Ca²⁺-ATPase, the same strategy was applied to the apparent K_Ca and h_Ca determining [Ca²⁺]_i-dependent activation. We also varied by ±2-fold the relative magnitudes of the reaction-kinetic constants k₁₂^o and k₂₁^o for the Ca²⁺-ATPase, while maintaining their ratio constant to avoid any thermodynamic bias. The constants k₁₂^o and k₂₁^o define the voltage dependence of the transport cycle, and their magnitudes relative to the reaction-kinetic constants for the rest of the cycle determines the range of voltages over which transport flux is voltage sensitive (Hansen et al., 1981). The fundamental parameters for the Ca²⁺ channel are constrained through direct experimental analysis to values well within this ±2-fold range (Hamilton et al., 2000, 2001; Pei et al., 2000; Köhler and Blatt, 2002; Supplemental Table S4), whereas this is not the case for the Ca²⁺-ATPase. So the comparison also offered a useful context for the parameterization of the pump. As expected, altering any of the parameters, either for Ca²⁺ channel or the Ca²⁺-ATPase, affected all macroscopic outputs as well as [Ca²⁺]_i, notably in the Open Reference State. Less obvious was their influence on cytosolic and vacuolar pH, most evident in each case in the Closed Reference State; this connection between Ca²⁺ and pH arises in part through their overlaps in [Ca²⁺]_i-mediated regulation of transport across the tonoplast, and we return to the relationship between [Ca²⁺]_i- and pH-dependent transport in the following article (Chen et al., 2012). These details aside, the similarities between pump and channel in scale and distribution of the outputs indicates a similar degree in sensitivity to variations in parameters for the Ca²⁺-ATPase.

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Figure 5.

Relative sensitivity of the OnGuard model to biophysical and kinetic parameters of the plasma membrane Ca²⁺ channel (A) and the tonoplast Ca²⁺-ATPase (B). Parameters were varied ±2-fold (for V_1/2, the near equivalent of ±18 mV) in each case and model outputs collected and normalized as in Figure 4. Additional details are provided in the text.

Testing Reference State Behavior with Environmental Challenge

Ultimately, the validity of any homeostatic model lies in its ability to recapitulate physiological behavior. There is much quantitative experimental detail relating to the physiological and macroscopic responses of guard cells to their ionic environment (see Supplemental Tables S1 and S2; Willmer and Fricker, 1996). Therefore, we varied external solute concentrations for both the closed and open reference states to compare the effects on a range of outputs. Simulations were run with KCl concentrations ranging from 1 to 30 mm, with CaCl₂ concentrations from 0.3 to 3 mm, and pH values from 6.0 to 7.0. The outputs demonstrated that the model successfully recapitulates stomatal behavior under each of these conditions. For example, increasing KCl concentration resulted in a progressive rise in guard cell volume and turgor in the open reference state, with stomatal aperture rising from around 8 μm in 1 mm KCl to almost 15 μm in 20 to 30 mm KCl (Fig. 6A). In the closed reference state, guard cell volume and turgor showed a slight decrease with increasing KCl concentration and stomatal aperture declined below 4 μm at the higher KCl concentrations, consistent with the increased osmotic load outside. Complementary and physiologically sensible outputs were evident for every other experimentally measurable variable, including membrane voltage (Fig. 6B), cytosolic and vacuolar K⁺, Cl⁻, and Mal concentrations (Fig. 7, A and B), pH_i, pH_v, and [Ca²⁺]_i (Fig. 8, A and B).

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Figure 6.

Response of the OnGuard model to environmental perturbations. Model outputs were determined in the Closed and Open Reference States with the standard environmental parameters of 10 mm KCl, 1 mm CaCl₂, and pH 6.5, and when each of these parameters was varied as indicated. Shown are the macroscopic outputs of stomatal aperture, guard cell volume and turgor (A), and plasma membrane and tonoplast voltage (B).

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Figure 7.

Response of the OnGuard model to environmental perturbations. Model outputs were determined in the Closed and Open Reference States as in Figure 6. Shown are the principle osmotic contents of [K⁺], [Cl⁻], and [Mal] in the cytosol (A) and the vacuole (B).

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Figure 8.

Response of the OnGuard model to environmental perturbations. Model outputs were determined in the Closed and Open Reference States as in Figure 6. Shown are the cytosolic and vacuole pH (A) and the cytosolic-free [Ca²⁺] ([Ca²⁺]_i).

These outputs also demonstrate a number of less obvious, but equally important trends, again recapitulating well-documented experimental data (Willmer and Fricker, 1996; see also Raschke and Schnabl, 1978; Van Kirk and Raschke, 1978; Blatt, 1987b, 2000; Lohse and Hedrich, 1992; Talbott and Zeiger, 1996; Dodd et al., 2005; Wang and Blatt, 2011). Among these, plasma membrane voltage showed a steeper dependence on KCl concentration in the Closed than in the Open Reference State (Fig. 6B); voltages in both states, and aperture in the Open Reference State, declined with CaCl₂ concentration (Fig. 6B) while [Ca²⁺]_i rose (Fig. 8B); increasing KCl concentration was accompanied by biphasic, and opposing changes in Cl⁻ and Mal in the vacuole in the Closed Reference State; and decreasing pH outside promoted cytosolic K⁺ concentration, vacuolar K⁺, and Mal concentrations (Fig. 7, A and B), and stomatal aperture (Fig. 6A) in the open reference state, but had little influence on membrane voltage, pH_i, or pH_v (Figs. 6B and 8A). Finally, we note that, in every case, [Ca²⁺]_i was elevated in the open compared with the Closed Reference State (Fig. 8B). The bases for these observations are fully explained by emergent interactions between the several transporters, the dominance of the plasma membrane H⁺-ATPase in the Open Reference State, and the influence of membrane voltage on transport, especially across the plasma membrane. We explore in depth these interactions, and their consequences, in the following article (Chen et al., 2012).