OnGuard, a Computational Platform for Quantitative - Plant Physiology

OnGuard, a Computational Platform for QuantitativeKinetic Modeling of Guard Cell Physiology1[W][OA]

Adrian Hills2, Zhong-Hua Chen2,3, Anna Amtmann, Michael R. Blatt4*, and Virgilio L. Lew4

Laboratory of Plant Physiology and Biophysics, University of Glasgow, Glasgow G12 8QQ, United Kingdom(A.H., Z.-H.C., A.A., M.R.B.); and Physiological Laboratory, University of Cambridge, Cambridge CB2 3EG,United Kingdom (V.L.L.)

Stomatal guard cells play a key role in gas exchange for photosynthesis while minimizing transpirational water loss from plantsby opening and closing the stomatal pore. Foliar gas exchange has long been incorporated into mathematical models, several ofwhich are robust enough to recapitulate transpirational characteristics at the whole-plant and community levels. Few models ofstomata have been developed from the bottom up, however, and none are sufficiently generalized to be widely applicable inpredicting stomatal behavior at a cellular level. We describe here the construction of computational models for the guard cell,building on the wealth of biophysical and kinetic knowledge available for guard cell transport, signaling, and homeostasis. TheOnGuard software was constructed with the HoTSig library to incorporate explicitly all of the fundamental properties fortransporters at the plasma membrane and tonoplast, the salient features of osmolite metabolism, and the major controls ofcytosolic-free Ca2+ concentration and pH. The library engenders a structured approach to tier and interrelate computationalelements, and the OnGuard software allows ready access to parameters and equations ‘on the fly’ while enabling the network ofcomponents within each model to interact computationally. We show that an OnGuard model readily achieves stability in a setof physiologically sensible baseline or Reference States; we also show the robustness of these Reference States in adjusting tochanges in environmental parameters and the activities of major groups of transporters both at the tonoplast and plasmamembrane. The following article addresses the predictive power of the OnGuard model to generate unexpected andcounterintuitive outputs.

Stomatal guard cells surround pores in the epider-mis of plant leaves and regulate the pore aperture.They open the pore in response to low CO2 and light tofacilitate CO2 access for photosynthesis, and they closethe pore in the dark, under drought stress, and in thepresence of the water-stress hormone abscisic acid tominimize water loss through transpiration. Stomatahave a profound impact on the water and carbon cy-cles of the world (Gedney et al., 2006; Betts et al., 2007).Their dynamics have been incorporated into modelsfor transpiration and water use efficiency (Farquharand Wong, 1984; Ball, 1987; Williams et al., 1996;

Eamus and Shanahan, 2002; West et al., 2005), suc-cessfully reproducing the gas exchange, CO2, andtranspirational characteristics of experiments at theplant and community levels. To date, these modelshave taken a top-down approach. They subsume sto-matal movements within a few empirical parametersof linear hydraulic pathways and conductances with-out reference to the molecular mechanics of the guardcell. No generalized guard cell model has yet to bedeveloped from the bottom up, drawing on the wealthof knowledge available for guard cell transport, sig-naling, and homeostasis. It is clear that such a model isnow needed. The depth and breadth of informationavailable for stomatal guard cells has made them thepremier cell system in plants for studies of membranetransport, signaling, and homeostasis; yet, in face ofthe complexity of the guard cell system, there remainsa very large gap in quantitative understanding abouthow guard cell transport works together to modulatesolute flux and regulate stomatal aperture.

A very large body of experimental evidence sup-ports the collective role of ionic fluxes across bothplasma membrane and tonoplast, and the metabolismof Suc and malate (Mal) in shaping the changes inosmotic load and turgor pressure that drive stomatalopening and closing (Willmer and Fricker, 1996;Blatt, 2000; Schroeder et al., 2001; Hetherington andWoodward, 2003; Sokolovski and Blatt, 2007). All of thepredominant transporters—the major K+, Cl2, and Mal-permeable channels and the H+-ATPases at the plasma

1 This work was supported by the UK Biotechnology and Biolog-ical Sciences Research Council (grant nos. BB/F001630/1, BB/F001673/1, and BB/H024867/1 to M.R.B.)

2 These authors contributed equally to the article.3 Present address: School of Science and Health, University of

Western Sydney, Hawkesbury Campus, Richmond, NSW 2753, Aus-tralia.

4 These authors contributed equally to the article.* Corresponding author; e-mail [email protected] author responsible for distribution of materials integral to the

findings presented in this article in accordance with the policy de-scribed in the Instructions for Authors (www.plantphysiol.org) is:Michael R. Blatt ([email protected]).

[W] The online version of this article contains Web-only data.[OA] Open Access articles can be viewed online without a subscrip-

tion.www.plantphysiol.org/cgi/doi/10.1104/pp.112.197244

1026 Plant Physiology�, July 2012, Vol. 159, pp. 1026–1042, www.plantphysiol.org � 2012 American Society of Plant Biologists. All Rights Reserved.

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membrane and tonoplast, as well as Ca2+-permeablechannels at both membranes (Allen and Sanders, 1997;MacRobbie, 1997; Sanders et al., 2002; White andBroadley, 2003; Dreyer et al., 2004; Peiter et al., 2005;Sokolovski and Blatt, 2007)—have each been isolatedand studied in sufficient depth to provide detailedand accurate flux equations with parameters fullyconstrained by experimental data. Much detail isavailable relevant to the activities of soluble enzymes,the metabolic pathways, and their kinetics that de-termine the synthesis, interconversion, and break-down of Suc and Mal within guard cells (Willmer andFricker, 1996).The information this body of data represents and

their nature poses a major challenge for its incorpora-tion and integration within any quantitative kineticmodel. Not only is there, for each transport process, aunique set of kinetic and regulatory descriptors, but forthe majority of transporters the process itself is inher-ently recursive, acting on one or more of these de-scriptors. For example, gating of the outward-rectifyingK+ channels is sensitive to membrane voltage as well asK+ concentration; depolarizing the membrane promotesthe current (Blatt, 1988b, 2000; Blatt and Gradmann,1997), yet the current generated on activating thesechannels normally draws K+ out of the cytosol, therebydriving the membrane voltage negative and sup-pressing channel gating. Indeed, each transport pro-cess carrying charge across a membrane affects—andis affected by—the voltage across that membrane, ifonly as a consequence of mass action and the move-ment of charged ions it carries. The problem is all themore complex because, for charge transporters op-erating across a common membrane, voltage is bothsubstrate and product and is shared between all ofthese charge-carrying transport processes (Weiss, 1996;Blatt, 2004a). Thus, the challenge becomes one of sys-tematically integrating each and every one of theseprocesses in a way that accommodates the recursivenature of transport and within an overarching strategythat is sufficiently flexible to allow parameter modifi-cations, even substitutions for the equations repre-senting each process, on the fly during experimentallyguided model refinements.To this end, we have expanded on an integrated

approach pioneered in models of mammalian epitheliaand erythrocytes (Lew et al., 1979, 2003; Lew andBookchin, 1986; Mauritz et al., 2009), incorporating inan iterative computational strategy the ensemble oftransport and buffering equations and their associatedvariables assigned by the operator. We describe theconcepts behind our development of the programminglibrary for Homeostasis, Transport and Signalling(HoTSig) and construction of the OnGuard softwarefor dynamic modeling of the guard cell. An importantfeature of the HoTSig library is its open structure,standardized with the major sets of equations anddescriptors for each of the various types of trans-porters. The OnGuard software incorporates a graph-ical user interface (GUI) that integrates with the

Microsoft Windows platforms and a Reference StateWizard for defining the starting point for in silico ex-perimentation. Here we show that OnGuard-generatedmodels readily achieve stability in a set of physiologi-cally relevant Reference States associated with the openand closed states of stomata, and we explore the subsetsof transporters and their parameters to which themodels are especially sensitive. We show that thesemodels are robust in adjusting to changes in envi-ronmental parameters. The predictive power of theOnGuard modeling approach is demonstrated withselected kinetic simulations in the companion article(Chen et al., 2012).

MODELING

The Modeling Strategy

Cellular homeostasis is especially well suited to anintegrative, bottom-upmathematical modeling approach.The physicochemical relations that constrain the behaviorof homeostatic variables—conservation of mass, charge,the electroneutrality of intracellular and extracellular so-lutions, the osmotic transport of water linked to all soluteconcentrations and their gradients, and of membranepotential linked to all ion gradients and permeabilities—are simple quantitative relations, all easy to model. Forthe guard cell, the most relevant homeostatic variablescomprise cell volume, cell osmolality, water potential andturgor, membrane potential, cytosolic and vacuolar K+,Cl2 and to lesser extents total and free Ca2+ contents andconcentrations, pH, unidirectional and net fluxes of allionic or neutral solutes through each transporter, wa-ter permeability, and flux. Also important from thestandpoint of transport and metabolic regulation arethe intracellular proton and Ca2+ buffering systems,the cell content of impermeant solutes, their osmoticcoefficients, charge, and its dependence on pH. Formost solutes, the lipid bilayer provides an effectivelyimpermeable barrier. Thus transport aside, the relationbetween the free and total concentrations of each sol-ute depends on factors such as buffering, macromo-lecular binding, and the degree of ionization. For all ofthe major solutes, quantitative data including buffer-ing constants are available or can be estimated forendogenous buffer systems as well as for all of theexperimentally applied buffers and chelators (e.g.HEPES, EGTA) in common use (Ferreira and Lew,1976; Tsien and Tsien, 1990; Föhr et al., 1993; Grabovand Blatt, 1997; Tiffert and Lew, 1997).

The biophysical relations of the different types oftransport that occur across eukaryotic membranes—mediated by pumps, channels, and carriers—are allwell understood and for the predominant transportershave been studied in sufficient depth to provide detailedand accurate flux equations (Weiss, 1996). The predom-inant ATP-driven ion pumps, H+-coupled transporters,and passive ion channels all have been characterizedwith regard to stoichiometry and mechanism, either in

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guard cells or in other plant cell types (Sanders, 1990;Willmer and Fricker, 1996; Blatt, 2004b), and their op-eration is readily described by sets of kinetic equations.Much of this kinetic detail applies directly to thetransporters in situ at the plasma membrane. Ourknowledge of transport across the tonoplast is less welldeveloped, largely because of difficulties to gain accessin vivo. In principle, the relative ignorance of transportat the tonoplast leaves quantitative modeling open toindetermination with unknown parameters. Nonethe-less, a large body of solid experimental results can betranslated as constraining equations on the system,including data on vacuolar ion contents and fluxes(MacRobbie, 1995, 2000, 2002; Willmer and Fricker,1996; Gobert et al., 2007) as well as the biophysicalconstraint of osmotic equilibrium between cytosol andvacuole. These constraints link the elemental transportand chemical processes to the macroscopic variables ofthe system such as the total guard cell volume (VT),turgor pressure (PT), and stomatal aperture (AS). (Acomplete list of abbreviations is provided in Supple-mental Appendix S5.) Thus, by careful design of theboundary equations it is possible to minimize therange of values for the set of unknown parameters thatcomplies with the macroscopic behavior of the system.This approach guided the design of the modelingreported here.

Key information is available also for the regulationof transport as well as that of Mal and Suc metabolism(Willmer and Fricker, 1996). Again, gaps exist in ourunderstanding of these controls but, from the contextof a modeler seeking to understand how the systemresponds to perturbation, the only relevant biology isencapsulated by how one model variable is connectedto another. It is frequently the case that this phenom-enology is directly accessible to experiment, whereasthe underlying mechanistic details are not, or are ac-cessible only qualitatively. For example, it is wellknown that an elevated [Ca2+]i inactivates IK,in of theguard cell, but we can only surmise that this may occurby its activation of a CBL-dependent protein kinase(Xu et al., 2006; Cheong et al., 2007) and we do nothave sufficient quantitative information to model ki-nase activation or K+ channel phosphorylation. Nev-ertheless, the quantitative relation between [Ca2+]i andK+ channel activity is known (Grabov and Blatt, 1999)and, by applying a mathematical description of thisrelation, we can safely place the mechanistic details ina black box that subsumes the intermediate kinetics.There are many other examples of the successful use ofblack-box phenomenology, including the Hodgkin-Huxley equations, which described the fundamentalphysiological processes of the Na+ and K+ channelsresponsible for action potentials in axons long beforethe underlying molecular mechanisms were elucidated(Hille, 2001). Such black boxes effectively parameterizephenomenological modules (Endy and Brent, 2001)and may be opened if, and when any elementssubsumed within a module become a target of the

modeling. This phenomenological approach also re-duces complexity and computational burden.

The HoTSig Library and OnGuard Software

By contrast with analytical approaches, numericalcomputation allows flexibility in modifying these rep-resentations during experimentally guided model re-finements. It is therefore important to resist temptationsto seek solutions that may appear to provide simplifiedor explicit analytical equations for voltage and the sumof membrane ion fluxes, but that sterilize the mainobjectives of the modeling exercise. We expanded onan iterative computational strategy applied suc-cessfully in the past (Lew et al., 1979, 2003; Lew andBookchin, 1986; Mauritz et al., 2009), developing a li-brary and software that incorporates the ensemble oftransport and buffering equations and their associatedvariables assigned by the operator. These equationswere included together with a set of macroscopic de-scriptors of guard-cell-specific features in constructionof the OnGuard software. In operation, the OnGuardsoftware calculated and logged the dynamic adjust-ments of ion flux and compartmental composition andof membrane voltage from a set of operator-definedstarting conditions; it used the sets of nonlinear dif-ferential flux equations for the transporters; and itobeyed the fundamental physical constraints of massand charge conservation. We built into the softwarethe facility to adjust the integration interval, takingaccount of the sums of each of the ionic fluxes suchthat the duration of iteration steps and the frequencyof data output could be set to adjust with the rate ofchange in the system (Lew and Bookchin, 1986; Lewet al., 1991), although in practice even short integrationperiods of 1 ms are handled with little loss of speedwhen run on a quad-processor-based computer typicalof those now widely available commercially. We pro-vide a brief explanation here. Further details of theHoTSig library and the OnGuard software are pro-vided in Supplemental Appendices S1 to S4.

Compartmental Analysis

The utility of a model relies on its facility for com-parison of its output with experimentally observedbehaviors. The vast majority of studies at the cellularlevel continue to be carried out on guard cells isolatedfrom the leaf in epidermals peels, or as protoplasts andisolated vacuoles, and maintained in controlled exter-nal media. As a starting point, we therefore treated theapoplast surrounding guard cells as an infinite reser-voir, uninfluenced by material entering or emanatingfrom the guard cell, leaving its composition to be de-fined by the operator as required for each particularsimulation. We also assumed a single endomembranecompartment, hereafter referred to as the vacuole. Thissimplification avoided the need to define additionalsets of poorly constrained transporters for other

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endomembrane compartments, although these may beadded later once sufficient experimental detail becomesavailable. Extracellular medium, guard cell cytosol, andvacuole therefore defined a semiclosed system of com-partments in series.

Preservation of Electroneutrality

The capacitative currents that charge the membraneare both transient and many orders of magnitudesmaller that those mediated by the ion fluxes relevantto cellular homeostasis (Findlay and Hope, 1976; Jacket al., 1983), and can thus be neglected for purposes ofmodeling here. Electroneutrality is preserved by theinstant value of the membrane potential across each ofthe membranes in the system, Vpm and Vton. These valuesensure that the sums of the individual ionic currentsthrough each of the electrogenic transporters on thetonoplast and plasma membrane are zero at all times.Thus, by setting the sum of all currents Ii at time t tozero, that is SIi(t) = 0 at each membrane, and solvingthese implicit equations for Vpm and Vton, respectively,we derived their values in each iteration, thereby en-suring that electroneutrality was preserved through-out all simulations. For each membrane, SIi(t) isnecessarily a complex function containing all of thedifferent kinetic representations of the various pump,channel, and carrier-mediated transporters.

Formulating the Constraining Equations

Stomatal dynamics are determined by the regulatedgain and loss of osmotically active solutes, Qi, withinthe guard cells and the associated changes in osmoticand turgor pressure (Raschke, 1979; MacRobbie andLettau, 1980; Willmer and Fricker, 1996). The compu-tational end product of the ensemble of all individualtransport and metabolic processes is the instantaneousvalue of the sum of all osmotically active soluteswithin the guard cell [SQi(T)], contained both withinvacuole [SQi(v)] and cytoplasm [SQi(c)]:

∑iQiðTÞ ¼ ∑

iQiðcÞ þ∑

iQiðvÞ ð1Þ

Two main constraints apply here, based on reliabledata (Hill and Findlay, 1981; Willmer and Fricker,1996): (1) the tonoplast cannot sustain hydrostaticpressure differences, and (2) the water permeabilityof both tonoplast and plasma membrane are high.Therefore, the osmotic pressure difference betweenvacuole and cytoplasm can be assumed to be zeroand the turgor pressure of the guard cell related solelyto the osmotic potential difference across the plasmamembrane. If, for simplicity, we rename QT = SQi(T),Qv = SQi(v), and Qc = SQi(c), and define VT, Vv,and Vc as the total volume of the guard cell, the vac-uole, and the cytosol compartments, respectively, thenthe new labels and the two constraints define theequalities:

Qv=Vv ¼ Qc=Vc ð2Þ

Vc=Vv ¼ Qc=Qv ð3Þ

QT ¼ QvþQc ð4Þ

VT ¼ Vvþ Vc; and hence ð5Þ

Qv=Vv ¼ Qc=Vc ¼ QT=VT ð6Þ

The osmotic pressure (P) at equilibrium with turgorpressure PT at each instant of time is given by the Van’tHoff equation:

P¼PT ¼ RT 3 �QT=VT2SCapo

� ð7Þ

where SCapo is the sum of the concentrations of allosmotically active solutes in the apoplast. Solving forVT yields:

VT ¼ QT=�SCapo þ PT=RT

� ð8Þ

Here, VT is linked with PT for each value of QT. QT inturn represents the end result of a large computationalsequence in each iteration of the model and reflects theosmotic load of the guard cell at that time. The twounknowns in Equation 8, VT and PT, will be complexfunctions of cell wall plasticity and its elastic modulus(Cosgrove, 1987). However experimental data linksthese variables empirically through the properties ofthe guard cell wall and the geometry of the stomata(Raschke, 1979; MacRobbie and Lettau, 1980; Willmerand Fricker, 1996; Franks et al., 2001) and satisfy therequirement for a constraining equation that describesthe link between VT and PT.

The relations between PT and stomatal aperture, AS,and between VT and AS have been measured for anumber of species, including Vicia, Commelina, tobacco(Nicotiana tabacum), and Arabidopsis (Arabidopsisthaliana; MacRobbie and Lettau, 1980; Franks et al.,2001; Shope et al., 2003; Gay, 2004; Meckel et al., 2007;Sokolovski et al., 2008). Both relations can be approx-imated by linear expressions with varying slopes andbaseline stomata closure values such that:

PT ¼ m 3 AS þ n ð9Þ

and

VT ¼ r 3 AS þ s ð10Þ

where m, n, r, and s are empirically determined con-stants.

From these equations, we can derive the desiredrelation between VT and PT using AS as a parametricvariable. Substituting from Equation 9 thus gives:



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AS ¼ PT=m2 n=m ð11Þ

and from Equation 10:

AS ¼ VT=r2 s=r : ð12Þ

Equating Equations 11 and 12 gives:

PT ¼ p 3 VT þ q ð13Þ

where p = m/r and q = n 2 m$s/r. Replacing PT fromEquation 13 in Equation 9 builds in the constraintsencoded by the experimentally determined Equations9 and 10, and enables a solution for VT and QT in eachcycle of computation. Replacing PT from Equation 8 inEquation 9 yields:

VT ¼ QT∑Capo þ ðpVT þ qÞ=RT ð14Þ

and rearranging gives:

pVT2

RTþ�Capo þ q

RT

�3VT2QT ¼ 0 ð15Þ

a quadratic equation in VT with the single solution:

VT ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi�Capo þ q=RT

�2 þ 4pQT=RTq

2Capo 2 q=RT

2p=RT

ð16Þ

With the values of QT and VT, the values of Vc, Vv, PT,and AS at any given time can be derived as:

Vv ¼ VT 3 Qv=QT; Vc ¼ VT 3 Qc=QT;

PT ¼ pVT þ q;

and AS ¼ ðVT2 sÞ=rð17Þ

In this strategy, all the particular geometrical featuresand wall properties of the guard cells are phenome-nologically encoded within the linear coefficients inEquations 9 to 13, allowing the operator to test species-related behavior based on the availability of experi-mental data that define these coefficients. This strategydelivers the value of all the macroscopic variables ofthe system (VT, PT, AS) at each instant of time, allow-ing comparisons between predicted and experimen-tally observed results. The critical micromacro link inthis strategy is the value of the osmotic load at eachinstant of time, QT(t), and its computation is the focusof the sections below.

Computing the Osmotic Load

The osmotic load, QT, of the guard cell responsiblefor stomatal dynamics arises from four main processes

comprising (1) membrane transport, (2) buffering re-actions, (3) metabolic synthesis and degradation reac-tions, and (4) signaling and regulatory reactions. Thereare many subgroups within each of these maincategories. Supplemental Tables S1 to S6 list the mac-roscopic constraints and the associated processesencoded in this initial OnGuard formulation. Details ofthe software implementation and additional explana-tions behind the choices in formulation will be foundin Supplemental Appendices S1 to S4 and Figs. S1-S3.Supplemental Tables S1 and S2 summarize the cel-lular and compartmental characteristics typical ofguard cells, with specific reference to those of Vicia.Supplemental Tables S3 to S6 outline the majortransmembrane ion transporters at the plasma mem-brane and tonoplast, along with their fundamentalbiophysical and regulatory characteristics, their basickinetic descriptors, and relevant literature. In each case,these transporters divide between primary, energy-driven ion pumps—the plasma membrane and vacuo-lar H+- and Ca2+-ATPases, and the vacuolar H+-PPase—H+-driven solute pumps such as H+-driven anionsymporters (Meharg and Blatt, 1995), and ion channelsincluding the slow vacuolar channel of the tonoplastidentified by the TPC1 gene of Arabidopsis (Peiteret al., 2005). Supplemental Appendix S1 includes de-scriptions of the buffering reactions for each com-partment relating to H+, Ca2+, and Mal, and theencapsulated metabolic reactions for Mal and Suc.The sequential steps followed in the computation ofQT(t) for each iteration during simulations are illus-trated in the flow diagram of Figure 1.

The User Interface

The operational core of the OnGuard softwarecomprises a GUI with a set of real-time displays of thestandard, steady-state current-voltage (IV) curves ateach membrane, tabular flux data, and a chart re-corder. The software incorporates typical, multiple-document interface construction, with each documentsaved to disk representing a complete OnGuardmodel. The GUI is built on the Microsoft Windowsplatform and Microsoft Foundations Classes and iswritten in the C++ programming language and pro-vides extensive, contextual help functionality. Duringsimulations, IV displays provide visual representa-tions of the kinetic characteristics for each of the vari-ous charge-carrying transporters in a scalable formatsimilar to that of the Henry Electrophysiology Suite(Hills and Volkov, 2004). Current-voltage curves areupdated in real-time along with the free-runningmembrane voltages at intervals specified by the oper-ator. These IV representations offer immediate feed-back on the evolving kinetic and thermodynamicfeatures for each of the transporters and they enablequantitative assessment of the underlying causal rela-tionships determining the temporal behavior of eachion flux. Illustrations of the IV displays are shown inFigure 2A and included with the results in the following


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article (Chen et al., 2012). Also shown, the tabular datasummarizes the net fluxes of each ionic species at bothmembranes as well as the contents of each ionic speciesin the three compartments, vacuole, cytosol, and apo-plast (Fig. 2B). Again, these data are available in realtime and, optionally, can be logged separately incomma-delimited format that can be read by spread-sheet programs such as Microsoft Excel and SigmaPlot.Finally, OnGuard provides a running graphical display—an on-screen chart recorder—of the most important,user-selectable data (Fig. 2C).The parameters of any model are accessed by means

of a series of property pages, drop-down lists, andpop-up dialogue boxes (Fig. 3A). These pages includegeneral buffering and environmental parameters suchas turgor pressure/aperture relations and ion concen-trations, the parameters specifying the various trans-porters at each membrane, parameters defining thecharacteristics for Suc and Mal metabolism, the light:dark cycle, and the time. Dialogue boxes for eachtransporter include operator-selectable controls thatdefine the inherent biophysical properties of the

transporter. For example, these controls enable accessto the voltage dependence of the inward-rectifying K+

channel and its gating charge, its ion selectivity andconductance, as well as options to define regulatoryparameters such as the kinetics for its inactivation by[Ca2+]i and its activation by extracellular H+ (Schroederet al., 1987; Blatt et al., 1990b; Blatt, 1992; Thiel et al.,1993; Grabov and Blatt, 1997, 1999); similarly, for thevarious pumps and ion-driven transporters thesecontrols provide access to the (pseudo-) rate constantsfor the carrier model, and the facility to define allo-steric (regulatory) ligand and light sensitivities. Fortransporters that do not carry charge and for transportactivities that are less well defined with respectto voltage or related kinetic properties—notablyH+-driven ion-exchange activities—simple, concentra-tion-driven transport definitions give access to trans-port stoichiometries, maximum transport rates, andapparent K1/2 values as well as any allosteric ligand orother regulatory sensitivities.

RESULTS

Model Construction with OnGuard

The usual approach to formulating dynamic modelsof cellular homeostasis begins with the definition of aninitial state—the Reference State—representing aphysiological resting condition of the system understudy (compare with Lew et al., 1979; Lew andBookchin, 1986). Once a Reference State is established,the operator introduces one or more perturbations thatrepresent new physiological, pathological, or experi-mental conditions to be explored and follows the re-sponse of all system variables as they evolve over time.

The Reference State Wizard

To establish a Reference State, we devised a softwarewizard that allows the operator to specify the under-lying biophysical status of the system—standard ionconcentrations in each compartment and voltagesacross the plasma membrane and tonoplast—and thenquery the model for net charge, driver ion (for plants,H+), and solute fluxes. The wizard allows the operatorto review and edit parameters such as membranevoltage and compartment solute composition, and todetermine parameters for metabolic equilibrium in Sucand Mal synthesis and catabolism; most important, thewizard gives access to pages that compare the fluxes ofionic species across each membrane and permit theirbalancing by adjustment to the population(s) of trans-porters and, as necessary, to their underlying kineticdescriptors (see Fig. 3B). Thus, the operator is able tointerrogate and adjust the subsets of transporters af-fecting the flux of each ion to satisfy the requirements ofa Reference State.

Obviously such manipulations imply a knowledgeof the (likely) unit density and/or limiting current

Figure 1. Computational flow of the OnGuard software. Computa-tional steps are summarized in the text and described in detail inSupplemental Appendix S1.



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amplitudes and their kinetic parameters for eachtransporter against which the operator can base judg-ment of biological validity. For most transporters of

guard cells this information is available, often towithin a factor of three and in many cases with anaccuracy of a few percent of the mean for the

Figure 2. Screenshots of the OnGuard user interface. A, The main window with current-voltage curve outputs relating totransport at the plasma membrane (left) and tonoplast (right). B, The tabular output window detailing the ionic and organicsolute contents within each compartment, the fluxes across the plasma membrane and tonoplast, the respective membranevoltages, the macroscopic outputs of cell volume, turgor and stomatal aperture, and the elapsed time counter. C, The graphicalchart-recorder output window with tab-selectable displays for each of the ionic and organic solute constituents of the cytosoland vacuole, pHi, pHv, [Ca

2+]i, and the respective membrane voltages (shown).


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parameter (see Supplemental Tables S3–S6). Evenwhen information for some transporters was uncertain,we found their characteristics sufficiently constrainedby the properties of other processes to ensure minimalindetermination. For example, with the H+-ATPase asthe sole pathway for H+ export across the plasmamembrane, achieving charge and net H+

flux balancein the Reference State required a close match with H+

consumption through Mal catabolism and with thedominant H+ return pathways, and required similar bal-ancing of the associated ion fluxes. Defining the H+-Cl2

and H+-K+ symporters included in our model, but forwhich there is limited information in guard cells (seebelow), required coordinating their relative contribu-tions to this H+ balance. The H+-coupling ratios of thesetransporters (H+-Cl2 symport returns two H+ per chargewhile H+-K+ symport returns 0.5 H+ per charge; Sanderset al., 1985; Blatt and Slayman, 1987) implies a 2:1 ratioin the activities of the two currents at the free-runningmembrane voltage to achieve charge balance in 1:1ratio with H+ export via the H+-ATPase. Additionally,H+-coupled Cl2 influx required an equal efflux of Cl2

through the sum of the anion channels, and H+-cou-pled K+ influx required an equal efflux of K+ throughthe outward-rectifying K+ channels. Finally, each of thesecomponents added a new current and therefore requiredbalancing with opposing currents of the H+-ATPase andK+ channels such that the total membrane current iszero at the free-running voltage. The consequence wasto constrain each of the currents to within a narrowrange of values relative to one another and within theconstraints of the known densities and current ampli-tudes for the H+-ATPase, K+, and Cl2 channels char-acteristic of the guard cells (see Supplemental TablesS3 and S4).

In practice, a systematic approach dictated that fluxesbe balanced first at the tonoplast and then at the plasmamembrane. Flux adjustment at each membrane beganwith nondriver ions (K+, Cl2, Ca2+, Mal22, Suc) andculminated with fine adjustments to the H+-ATPases(and for the tonoplast, the H+-PPase) densities. Trans-porter numbers were assessed against the available ex-perimental data and the process was repeated if theapproximation failed. We found it sufficient to bring allof the fluxes of each of the transported species in bal-ance to within 610215 mol s21; thereafter trial simula-tions achieved a stable Reference State, generally within8 to 10 h of simulation time, and maintained this con-dition even when allowed to run free over periodsequivalent to many months.

The Choice of Transporters and Their Parameters

Most of the relevant quantitative kinetic informationavailable comes from studies of the guard cells of Vicia

Figure 3. Screenshots of selected OnGuard property pages. A, Samplepages for access to the biophysical and kinetic parameters for each of

the various transporters, Suc and Mal metabolism, Ca2+ and H+ buff-ering. B, Selected pages within the Reference State Wizard.



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(Supplemental Tables S3–S6). Therefore, we used thiscell system as our starting point for model construc-tion. Nonetheless, a complete set of parameters for allof the pumps, carriers, and channels is not available forany one species. A comparison of transport parameterslisted in the tables shows a substantial degree of sim-ilarity between species, and even between cell types.For example, the fundamental gating characteristicsfor the plasma membrane K+ channels of Vicia, to-bacco, and Arabidopsis guard cells (see SupplementalTable S4) include overlapping values for gating charge(d) and half-activation voltage (V1/2), as well as asimilar sensitivities to extracellular [K+], cytosolic pH,and [Ca2+]i (Dreyer and Blatt, 2009). Furthermore,Supplemental Table S3 highlights the close quantita-tive relationships between the plasma membraneH+-ATPases of Vicia guard cells (Blatt, 1987a, 1988a),the giant alga Chara (Blatt et al., 1990a), and the fungusNeurospora (Gradmann et al., 1978; Sanders and Slayman,1982; Blatt and Slayman, 1987). For these reasons we feltconfident to borrow characteristics from other species—even other cell types—when information was lacking forVicia, as necessary scaling the transporter for Viciaguard cells. For example, kinetic detail of H+-coupledK+ symport is missing for guard cells. Nonetheless, anestimate of the probable symport current can be de-rived from measurements of K+ uptake against itselectrochemical gradient in Vicia guard cells (seeSupplemental Table S3; Blatt and Clint, 1989; Clint andBlatt, 1989) and a current similar to that documentedfor the H+-K+ symport of Arabidopsis roots (Maathuisand Sanders, 1994). We could have modeled thistransporter as a simple current source—that is, inde-pendent of membrane voltage—but we reasoned thatthe kinetic constraint of voltage was likely to be im-portant for the dynamics of K+ and H+ balance, muchas has been demonstrated in Neurospora (Blatt andSlayman, 1987). H+-coupled K+ transport was origi-nally described in Neurospora (Rodriguez-Navarroet al., 1986; Blatt and Slayman, 1987; Blatt et al., 1987)and, although the current is typically 10-fold greater inK+-starved Neurospora than in the plant cells, thequalitative kinetic dependencies on external [K+],pH, and voltage are similar. A relative weighting ofreaction-kinetic constants is available for the carriercycle of the H+-coupled K+ symport in Neurospora(Blatt et al., 1987); hence, we used this weighting andthe current amplitudes from Arabidopsis and esti-mated for Vicia as a guide in assigning values to thereaction-kinetic constants for the H+-K+ symport in theguard cells. Much less kinetic information is availablefor plasma membrane Ca2+-ATPases. Again, it wasunrealistic to assume its voltage independence: Bestestimates place the equilibrium voltage for the plasmamembrane Ca2+ pumps near 2200 mV with 1 mM [Ca2+]outside and, hence, within the physiological voltagerange. Instead, we borrowed the carrier cycle estab-lished for the H+-ATPase, scaling it to estimates of thetypical Ca2+ flux and measured values for the Km withrespect to [Ca2+]i, thereby ascribing a significant

dependence on membrane voltage over much of thephysiological range. Finally, to avoid some uncer-tainties in defining parameters we concatenatedtransport activities associated with Cl2 and NO3

2,subsuming NO3

2 with Cl2 as a single, anionic species.Both anions contribute to the osmotic content of theguard cells, their relative presence depending to alarge extent on availability, both are permeablethrough several of the anion channels present at thetwo membranes, and coupled transport for Cl2 andNO3

2 show similar thermodynamic and kinetic prop-erties, to the extent that they are known (Sanders et al.,1989; Meharg and Blatt, 1995). By this maneuver wecould draw on the available kinetic detail for H+-coupledNO3

2 transport at both membranes (SupplementalTables S3 and S5) and avoid redundancy with less-well-defined characteristics, notably for Cl2 transport at theplasma membrane.

A case-by-case summary of the reasoning behindthe selection of each transporter will be found inSupplemental Appendix S2. The complete set ofOnGuard parameters used for the modeling describedin this article will be found in Supplemental AppendixS6 and is available for download with a demonstrationof the OnGuard software (see www.psrg.org.uk). Ourinclusive approach taken in developing the modeldescribed below expands the number of model pa-rameters and, correspondingly, the model complexity.It is often a choice in modeling to use a skeleton con-struction to determine the minimum number of com-ponents, and associated parameters, needed tosimulate a particular set of biological phenomena. Inthis case, however, the parameter space for the en-semble of transporters is exceptionally well con-strained experimentally, generally to within a factor ofthree and frequently with an accuracy of a few percent(see Supplemental Tables S3–S6). This information,and the fundamental requirements for charge andionic balance, placed narrow limits on the character-istics of the remaining transporters, even when theirparameterization was less certain. As we demonstratehere, and in the companion article (Chen et al., 2012), adeterministic solution is readily found that recapitu-lates a very wide range of physiological behaviors.Nonetheless, the OnGuard software enables the user togenerate and test skeleton models with equal ease.

Closed and Open Reference States

To test the parameter space of OnGuard modelsand their robustness, we used the results of initialsimulations as a guide in defining a pair of states ofthe guard cell associated with the closed and openstomata, namely the Closed and Open ReferenceStates. The Closed Reference State was envisaged astypical of stomata in the dark, in which the guard cellsretained a baseline of osmotic load and a minimum ofion flux across the tonoplast and plasma membrane. Forpurposes of modeling, the currents of all primarypumps (H+-ATPases, Ca2+-ATPases, H+-PPase) at the


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tonoplast and plasma membrane were assigned valuesof 5%, 10%, or 20% of their outputs in the Open Refer-ence State, consistent with experimental estimates of theknown light-stimulated activities of plasma membraneATPases (Assmann et al., 1985; Shimazaki et al., 1986;Serrano et al., 1988; Goh et al., 1995, 1996; Kinoshitaet al., 2001; Chen et al., 2012). Again, for the ClosedReference State we sought parameter sets such that thenet fluxes of all solutes, Fi, across both tonoplast andplasma membrane were zero, that is SFi = 0 for eachmembrane. We then used the same model parame-ters for the Open Reference State, allowing only thestep up in primary pump currents, Suc and Malsynthesis. All other model variables were kept con-stant between these paired Reference States, and fineadjustments to the properties of individual trans-porters were introduced between simulations. Thus,Reference State pairing offered a convenient test ofthe capacity of the parameter ensemble to supportstomatal movement across a range of common en-vironmental variables.

As an initial test of the parameter ensemble, weexplored model robustness by varying systematicallythe surface densities of individual transporters withinbounds of 61.3-, 2-, and 5-fold of their starting valuesas dictated by experimental knowledge and uncer-tainty. Figure 4 summarizes the results of simulationswithin the 61.3-fold boundary reporting both macro-scopic (aperture, osmotic content, total Ca2+) and mi-croscopic (pHi, pHv, [Ca

2+]i) outputs. Displacements ofeach output were normalized to the correspondingvalue obtained with the starting parameters for pur-poses of comparison. In general, we found the modelto be relatively insensitive to several of the predomi-nant osmotic solute transporters at the plasma mem-brane. For example, a 1.3-fold increase or decrease inthe 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 oropen states. Only outside the physiological limits forthe channel population (see Supplemental Table S4)were variations in output returned that, in principle,might be detectable through biological experimenta-tion (not shown). Similar sensitivities were recoveredwith variations in densities for the outward-rectifyingK+ channel, and H+-Mal symport at the plasmamembrane, and for analogs of the TPK1, TPC1, FV,VCL, and VMAL channels at the tonoplast. These re-sults demonstrated a considerable robustness despitethe corresponding ranges in densities associated with

Figure 4. Relative sensitivity of the OnGuard model to the com-ponent transport activities at the plasma membrane (A and B) andtonoplast (C and D) when transporter densities were varied 61.3-

fold about the starting values. Model outputs (ni) were determined inthe Closed (A and C) and Open (B and D) Reference States and nor-malized to the outputs with the corresponding starting values (no). Forpurposes of comparison, outputs are reported for selected macroscopic(stomatal aperture, total osmotic content, and total Ca2+ content) andmicroscopic ([H+]i, [H

+]v, and [Ca2+]i) variables. Additional details areprovided in the text.



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these transporters. They also suggest a substantialfunctional tolerance for divergent transporter densitiesin the guard cell.

The OnGuard model proved substantially moresensitive to variations in the densities of transportersdirectly affecting [Ca2+]i and cytosolic pH (see Fig. 4).As central signaling and energetic elements, both [Ca2+]iand cytosolic pH affect an extended network oftransporters at both membranes (Blatt, 2000; Schroederet al., 2001) and, thus, were anticipated to imposegreater restrictions on performance. Substantiallygreater relative variations in macro- and/or micro-scopic outputs were recovered, even with 1.3-foldchanges in densities of the H+-ATPase, Ca2+-ATPase,and Ca2+ channel at the plasma membrane, and for theCa2+-ATPase and Ca2+ channel at the tonoplast. No-table sensitivities, especially in cytosolic and/or vac-uolar [H+] were evident also to changes in the densitiesof the H+-Cl2 and H+-K+ symporters at the plasmamembrane and the CLC, NHX, and CAX antiporters atthe tonoplast. Aperture and osmotic content sensitiv-ities to H+-Cl2 symport density is consistent with itscentral role in anion uptake and, for reasons notedabove, variations in either the H+-Cl2 symport, H+-K+

symport, or H+-ATPase densities had the greatest im-pact in the Closed Reference State, especially on pHibalance. The roles for the endosomal cation and anionantiporters in regulating pH are broadly consistentwith observation documented in the literature (Pardoet 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 thebiophysical and kinetic parameters for transporterswith appreciable effects on both the macro- and mi-croscopic parameters in Figure 4. As examples, Figure5 summarizes the outputs from simulations with theplasma membrane Ca2+ channel and the tonoplastCa2+-ATPase, both of which contribute to the Ca2+

circuit of the model. For the Ca2+ channel, we variedby 62-fold each of the key gating parameters affectingits voltage sensitivity—V1/2 and d that define thevoltage-yielding half-maximal conductance and thesensitivity of the channel gate to changes in voltage,respectively—and the apparent KCa and associated Hillcoefficient, hCa, for channel inactivation by [Ca2+]i. Forthe Ca2+-ATPase, the same strategy was applied to theapparent KCa and hCa determining [Ca2+]i-dependentactivation. We also varied by 62-fold the relativemagnitudes of the reaction-kinetic constants k12

o andk21

o for the Ca2+-ATPase, while maintaining their ratioconstant to avoid any thermodynamic bias. The con-stants k12

o and k21o define the voltage dependence of

the transport cycle, and their magnitudes relative tothe reaction-kinetic constants for the rest of the cycledetermines the range of voltages over which transportflux is voltage sensitive (Hansen et al., 1981). Thefundamental parameters for the Ca2+ channel areconstrained through direct experimental analysis tovalues well within this 62-fold range (Hamilton et al.,2000, 2001; Pei et al., 2000; Köhler and Blatt, 2002;

Supplemental Table S4), whereas this is not the case forthe Ca2+-ATPase. So the comparison also offered auseful context for the parameterization of the pump.As expected, altering any of the parameters, eitherfor Ca2+ channel or the Ca2+-ATPase, affected allmacroscopic outputs as well as [Ca2+]i, notably in theOpen Reference State. Less obvious was their influenceon cytosolic and vacuolar pH, most evident in eachcase in the Closed Reference State; this connectionbetween Ca2+ and pH arises in part through theiroverlaps in [Ca2+]i-mediated regulation of transportacross the tonoplast, and we return to the relationshipbetween [Ca2+]i- and pH-dependent transport in thefollowing article (Chen et al., 2012). These details aside,the similarities between pump and channel in scaleand distribution of the outputs indicates a similar de-gree in sensitivity to variations in parameters for theCa2+-ATPase.

Testing Reference State Behavior with Environmental Challenge

Ultimately, the validity of any homeostatic modellies in its ability to recapitulate physiological behavior.There is much quantitative experimental detail relating

Figure 5. Relative sensitivity of the OnGuard model to biophysicaland kinetic parameters of the plasma membrane Ca2+ channel (A) andthe tonoplast Ca2+-ATPase (B). Parameters were varied 62-fold (for V1/2,the near equivalent of 618 mV) in each case and model outputscollected and normalized as in Figure 4. Additional details are pro-vided in the text.


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to the physiological and macroscopic responses of guardcells to their ionic environment (see Supplemental TablesS1 and S2; Willmer and Fricker, 1996). Therefore, wevaried external solute concentrations for both the closedand open reference states to compare the effects on arange of outputs. Simulations were run with KCl con-centrations ranging from 1 to 30 mM, with CaCl2 con-centrations from 0.3 to 3 mM, and pH values from 6.0 to7.0. The outputs demonstrated that the model success-fully recapitulates stomatal behavior under each of theseconditions. For example, increasing KCl concentrationresulted in a progressive rise in guard cell volume andturgor in the open reference state, with stomatal aperture

rising from around 8 mm in 1 mM KCl to almost 15 mm in20 to 30 mM KCl (Fig. 6A). In the closed reference state,guard cell volume and turgor showed a slight decreasewith increasing KCl concentration and stomatal aperturedeclined below 4 mm at the higher KCl concentrations,consistent with the increased osmotic load outside.Complementary and physiologically sensible outputswere evident for every other experimentally measurablevariable, including membrane voltage (Fig. 6B), cytosolicand vacuolar K+, Cl2, and Mal concentrations (Fig. 7, Aand B), pHi, pHv, and [Ca2+]i (Fig. 8, A and B).

These outputs also demonstrate a number of less ob-vious, but equally important trends, again recapitulatingwell-documented experimental data (Willmer and Fricker,1996; see also Raschke and Schnabl, 1978; Van Kirk andRaschke, 1978; Blatt, 1987b, 2000; Lohse and Hedrich,1992; Talbott and Zeiger, 1996; Dodd et al., 2005; Wangand Blatt, 2011). Among these, plasma membrane voltageshowed a steeper dependence on KCl concentration in theClosed than in the Open Reference State (Fig. 6B); voltagesin both states, and aperture in the Open Reference State,declined with CaCl2 concentration (Fig. 6B) while [Ca2+]irose (Fig. 8B); increasing KCl concentration was accom-panied by biphasic, and opposing changes in Cl2 andMalin the vacuole in the Closed Reference State; and de-creasing pH outside promoted cytosolic K+ concentration,vacuolar K+, andMal concentrations (Fig. 7, A and B), andstomatal aperture (Fig. 6A) in the open reference state, buthad little influence on membrane voltage, pHi, or pHv(Figs. 6B and 8A). Finally, we note that, in every case,[Ca2+]i was elevated in the open compared with theClosed Reference State (Fig. 8B). The bases for these ob-servations are fully explained by emergent interactionsbetween the several transporters, the dominance of theplasma membrane H+-ATPase in the Open ReferenceState, and the influence of membrane voltage on trans-port, especially across the plasma membrane. We explorein depth these interactions, and their consequences, in thefollowing article (Chen et al., 2012).

DISCUSSION

A major challenge in constructing any quantitative ki-netic model of transport and homeostasis is to integratethe often substantial body of data in a systematic repre-sentation of the cell. Although basic physicochemical re-lationships are all simple quantitative relations easilyincorporated in a quantitative description of transport, therecursive nature of charge transport presents a challeng-ing dimension to any modeling effort that generally defiessimple analytical solution. We have developed an openstructured approach to software construction that shouldprove widely applicable to modeling cellular transportand homeostasis. HoTSig provides a standardized librarythat incorporates all of the major sets of equations anddescriptors for transport across biological membranes,and a minimum set of equations defining the metabolismof the major organic solutes. The OnGuard software,constructed from this library, incorporates a GUI for real-

Figure 6. Response of the OnGuard model to environmental pertur-bations. Model outputs were determined in the Closed and OpenReference States with the standard environmental parameters of 10 mM

KCl, 1 mM CaCl2, and pH 6.5, and when each of these parameters wasvaried as indicated. Shown are the macroscopic outputs of stomatalaperture, guard cell volume and turgor (A), and plasma membrane andtonoplast voltage (B).



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time feedback on the individual transport and homeo-static processes to model the physiological behavior ofstomatal guard cells; it incorporates a set of empiricallydefined equations to relate the output of solute content tocell volume, turgor, and stomatal aperture. Additionally,we introduce the Reference State Wizard and outline itsuse in defining a starting point for in silico experimenta-tion with sensible outputs for all known variables. Finally,we show that an OnGuard model, with a realistic en-semble of transporters, is able to recapitulate the knowncharacteristics of guard cells and stomata in the face ofcommon experimental manipulations.

A Quantitative Modeling Approach

The few instances in which mathematical modelinghas been applied to cellular homeostasis with suffi-cient rigor have been remarkably successful both in

reproducing known cellular physiology and in pre-dicting unexpected behaviors. Even so, the subject isoften perceived as difficult or inaccessible. Mostmodeling efforts have been implemented on a case-by-case basis without a standardized format and, as aconsequence, mastering all of the individual metabolic,transport, and buffering mechanisms presents a chal-lenge, especially as much of the quantitative data re-lating to membrane transport are to be found in olderliterature and in formats that are not readily accessibleexcept to the specialist. Utilities such as the Virtual Cell(Loew and Schaff, 2001) provide for modeling intracel-lular events that encompass reaction-diffusion processesin arbitrary geometries, but offer limited flexibility indefining underlying behaviors, for example as dictatedby specific transport equations. These limitations areaddressed in our development of the HoTSig approach.

Figure 7. Response of the OnGuard model to environmental pertur-bations. Model outputs were determined in the Closed and OpenReference States as in Figure 6. Shown are the principle osmoticcontents of [K+], [Cl2], and [Mal] in the cytosol (A) and the vacuole (B).

Figure 8. Response of the OnGuard model to environmental pertur-bations. Model outputs were determined in the Closed and OpenReference States as in Figure 6. Shown are the cytosolic and vacuolepH (A) and the cytosolic-free [Ca2+] ([Ca2+]i).


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The software contains an expandable library for trans-porter kinetics, chemical buffering, and capacity formacromolecular binding and metabolic reactions, allaccessible to input and modification by the operator.This open structure should make HoTSig adaptable forwide variety of single-cell systems.The HoTSig library and OnGuard software greatly

expands on earlier efforts of our own (Lew et al., 1979;Lew and Bookchin, 1986; Lew, 1991; Gradmann et al.,1993) and others (Loew and Schaff, 2001; Hunter andNielsen, 2005; Shabala et al., 2006) in providing real-time feedback for the operator as well as a detailedoutput log. We adapted a number of generalized utili-ties that greatly reduce computational load and time,notably the Newton-Raphson chord approximation tosolutions of implicit equations (Lew and Bookchin,1986), and we introduced a look-ahead utility to adjusttime increments to the pending dynamics of the model(Supplemental Appendices S3 and S4). We have placedconsiderable emphasis on an intuitive GUI for the entryof initial conditions, simulated perturbations, and fortabular and graphical displays to make simulationoutputs readily apparent to the operator. Unlike theearlier efforts, the OnGuard graphical interface pro-vides detailed flux and kinetic information in tabularand chart-recorder formats as well as real-time electro-physiological displays with the component and en-semble current-voltage relations at each membrane. Weanticipate users of the OnGuard software will start withthe defaults specified by the parameter set available fordownload with the demonstration (see www.psrg.org.uk), thereafter customizing individual transportersand/or environmental conditions to address specificphysiological questions, species variations, and theconsequences of genetic manipulations. Interpreting thecurrent-voltage outputs requires some knowledge ofelectrophysiology; nonetheless, the tabular and chart-recorder formats offer useful reference points for com-parison in real time.To make this process as rapid and intuitive as possible,

each descriptor in a model is editable and parameters areaccessible on the fly during modeling sessions. As edit-able modules, these descriptors serve as phenomeno-logical black boxes to be opened, or reduced, wheneverthe internal workings become a desirable or necessarypart of a modeling project. From the point of view of amodeler seeking to understand how the homeostaticvariables of a given system respond to a physiological orexperimental perturbation, the only relevant biology isencapsulated by how one model variable is connected toanother. It is frequently the case that this phenomenol-ogy is directly accessible to experiment, whereas theunderlying mechanistic details are not, or are accessibleonly qualitatively. By fitting the experimental data with amathematical relation, they are safely placed within ablack box that represents a parameterized module withadjustable levels of resolution (Endy and Brent, 2001)and may be opened if, and when the included regulatorypathway becomes a target of study for the purposes ofmodeling. In effect, this phenomenological approach

serves as a place holder for unknowns and as a means toreduce complexity and computational burden.

We anticipate that the HoTSig library will find generalapplications in exploring cell systems, in addition toguard cells, for which sufficient biophysical and kineticdetail for transport is now available. For example, itshould prove useful in exploring the physiology of theplant root epidermis and its interaction with the soilenvironment. Root epidermal cells, including root hairs,function in the uptake of essential mineral nutrients fromthe soil, but also provide entry pathways for toxic ions,including heavy metals and Na+, that arise through in-tensive irrigation and pollution, and present a majorchallenge to modern agriculture. Aspects of Na+ toxicityas well as pH, Ca2+, and K+ transport oscillations havefound their way into computational assessments of iontransport in roots (Amtmann and Sanders, 1999; Shabalaet al., 2006), but will now benefit from quantitativeanalysis incorporating appropriate reference states thathelp minimize indetermination and validate predictivepower. Finally, the OnGuard software is equally appli-cable to guard cells of species other than Vicia. As wenoted above (see “The Choice of Transporters and TheirParameters”), there exists substantial quantitative simi-larity between the guard cells of a number of plantspecies, including Vicia, Nicotiana, and Arabidopsis. Inlarge measure, adapting the model we resolved for Viciato other species requires only an accounting for diff-erences in cell geometry and the relationships betweencell surface area, volume, and turgor pressure.

Modeling Guard Cell Dynamics

In practice, the OnGuard software returned anensemble of model parameters that, in simulations oflimiting stomatal behavior, were robust and reca-pitulated the predominant characteristics of guardcell physiology with a bare minimum of intrinsicassumptions. Once resolved with the Reference StateWizard, we found little difficulty in establishing amodel using parameters within experimentally de-termined constraints. Often Monte Carlo methodsare used to determine the best solution(s) to simulationsthat must deal with an extended, n-dimensional pa-rameter space when a deterministic solution cannotbe found (Moskowitz and Caflisch, 1996; Robertand Casella, 2011). However, for all but a few of theguard cell transporters, the parameter space is ex-ceptionally well constrained experimentally, gener-ally to within a factor of three and frequently with anaccuracy of a few percent (see Supplemental TablesS3–S6). This information, and the fundamental re-quirements for charge and ionic balance, placed narrowlimits on the characteristics of the remaining trans-porters, even when their parameterization was lesscertain.

We found this model recapitulated the guard cell inthe closed (dark) and open (light) states of the sto-mata with the single assumption of a defined, light-



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dependent increase in the activities of all ion-translocatingATPases and pyrophosphatase at the plasma mem-brane and tonoplast, and in Suc and Mal synthesis(Willmer and Fricker, 1996; Shimazaki et al., 2007).These paired reference states showed substantial in-trinsic stability in the face of systematic changes to arange of parameters, including transporter densities atboth membranes and the gating and regulatory char-acteristics of channels and pumps, the densities ofwhich had the greatest effect on model outputs (Figs. 4and 5). Perturbations in these parameters resulted pri-marily in differences in the dynamic range of macro-scopic outputs, on [Ca2+]i, pHi, and pHv. Nonetheless,these outputs compare favorably with the literature(see Supplemental Tables S1 and S2; Figs. 6–8;Raschke, 1979; MacRobbie and Lettau, 1980; Hill andFindlay, 1981; Zeiger, 1983; Davies and Jones, 1991;Willmer and Fricker, 1996; Blatt, 2000; Schroeder et al.,2001; Shimazaki et al., 2007). Furthermore, the modelsuccessfully recapitulated stomatal characteristics (seeFigs. 6–8) over a wide range of extracellular ion con-centrations and pH for which there is substantial ex-perimental documentation. These results fulfill a setof minimum criteria in validating the model andthey also offer unexpected perspectives on the conse-quences of these manipulations, especially in the re-lationship between [Ca2+]i, cytosolic, and vacuolar pH.In the following article (Chen et al., 2012) we reviewthese, and additional interactions, underlining thepower of the OnGuard model in providing (other-wise counterintuitive) explanations for publishedexperimental data with a minimum of underlyingassumptions.

MATERIALS AND METHODSDetails of the model assembly and computational components will be found

in the Supplemental Appendices S1 to S4.

Supplemental Data

The following materials are available in the online version of this article.

Supplemental Figure S1. Generalized four-state reaction kinetic cycle.

Supplemental Figure S2. Malic acid (de-) protonation reactions.

Supplemental Figure S3. Global metabolic reactions for synthesis and deg-radation of sucrose and malic acid.

Supplemental Table S1. Basic biophysical parameters of Vicia stomatalguard cells.

Supplemental Table S2. Compartmental contents of Vicia stomatal guardcells.

Supplemental Table S3. Predominant plasma membrane pumps and car-riers.

Supplemental Table S4. Predominant plasma membrane ion channels.

Supplemental Table S5. Predominant tonoplast pumps and carriers.

Supplemental Table S6. Predominant tonoplast channels.

Supplemental Appendix S1. The HoTSig Platform and OnGuard software.

Supplemental Appendix S2. A brief summary of transporter selections.

Supplemental Appendix S3. Implementing Newton-Raphson approxima-tions.

Supplemental Appendix S4. Determing Dt.

Supplemental Appendix S5. Abbreviations.

Supplemental Appendix S6. RCF1 model parameters.

ACKNOWLEDGMENTS

We thank Simon Rogers,Yizhou Wang, and Christopher Grefen (Universityof Glasgow) for comments during the development of the software and model.

Received March 16, 2012; accepted May 20, 2012; published May 25, 2012.

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