-
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 qRT
�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 are
provided 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 [Ca
2+]i. Forthe Ca2+-ATPase, the same strategy was applied to
theapparent KCa and hCa determining [Ca
2+]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 [Ca
2+]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
[Ca
2+]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 mMKCl, 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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