A Dynamic Mathematical Model To Clarify Signaling Circuitry Underlying Programmed Cell Death Control...

A Dynamic Mathematical Model To Clarify Signaling CircuitryUnderlying Programmed Cell Death Control in Arabidopsis DiseaseResistance

Vikas Agrawal,† Chu Zhang,† Allan D. Shapiro,*,† and Prasad S. Dhurjati*,‡

Department of Plant and Soil Sciences, Delaware Agricultural Experiment Station, College of Agriculture andNatural Resources, and Department of Chemical Engineering, College of Engineering, University of Delaware,Newark, Delaware 19716

Plant cells undergo programmed cell death in response to invading pathogens. Thiscell death limits the spread of the infection and triggers whole plant antimicrobialand immune responses. The signaling network connecting molecular recognition ofpathogens to these responses is a prime target for manipulation in genetic engineeringstrategies designed to improve crop plant disease resistance. Moreover, as alterationsto metabolism can be misinterpreted as pathogen infection, successful plant metabolicengineering will ultimately depend on controlling these signaling pathways to avoidinadvertent activation of cell death. Programmed cell death resulting from infectionof Arabidopsis thaliana with Pseudomonas syringae bacterial pathogens was chosenas a model system. Signaling circuitry hypotheses in this model system were testedby construction of a differential-equations-based mathematical model. Model-basedsimulations of time evolution of signaling components matched experimental measure-ments of programmed cell death and associated signaling components obtained in acompanion study. Simulation of systems-level consequences of mutations used inlaboratory studies led to two major improvements in understanding of signalingcircuitry: (1) Simulations supported experimental evidence that a negative feedbackloop in salicylic acid biosynthesis postulated by others does not exist. (2) Simulationsshowed that a second negative regulatory circuit for which there was strongexperimental support did not affect one of two pathways leading to programmed celldeath. Simulations also generated testable predictions to guide future experiments.Additional testable hypotheses were generated by results of individually varying eachmodel parameter over 2 orders of magnitude that predicted biologically importantchanges to system dynamics. These predictions will be tested in future laboratorystudies designed to further elucidate the signaling network control structure.

Introduction

All living organisms mount an active defense againstpotential pathogens. Defense against pathogenic mi-crobes typically involves either direct molecular recogni-tion of microbial components or indirect perception ofaltered host metabolism (1). Recognition leads to activa-tion of antimicrobial and immune responses (2). One ofthese defense responses to plant pathogens is highlylocalized programmed cell death (PCD) of plant cells inthe infected region. This localized PCD is assumed to helpcontain pathogen spread (3) and is known to triggersystemic immune responses (4, 5). PCD has similarlybeen shown to contribute to antibacterial defense innematodes (6). By contrast, some bacterial pathogens ofmammals exploit PCD to cripple the immune system (7-13). Control over PCD can thus be central to determiningsusceptibility or resistance to disease.

Plant diseases cause annual worldwide losses of plantproductivity valued at over $100 billion (14, 15). Croplosses routinely lead to hunger and malnourishment inthe developing world. Manipulation of the PCD is apromising strategy for engineering disease control (16,17). Moreover, there is increasing evidence that PCDcontrol pathways are important for plant metabolicengineering. A diverse array of genes with no obviousconnection to PCD or plant response to microbes havebeen shown to cause PCD when expressed as transgenes(18). Plants that show PCD in the absence of pathogen(lesion mimic mutants) are one of the largest classes ofmutants obtained from genetic screens (19-21). It ap-pears that perturbations to many aspects of plant me-tabolism and signaling cause the plant to respond as iffacing an infection (22-26). Avoiding inadvertent activa-tion of PCD is of obvious significance to genetic engineer-ing efforts that involve directed changes to metabolism.

PCD and a battery of other defense responses areinduced by a diverse array of microbial plant pathogensincluding viruses, bacteria, fungi, and oomycetes (27).Molecular recognition of these diverse pathogens isconnected to common downstream defense responses bya signal transduction network. The existence of this

* To whom correspondence should be addressed. A.D.S.: Tel.(302) 831-4889. Fax (302) 831-3447. Email [email protected].: Tel (302) 831-2879. Fax (302) 831-1048. Email [email protected].

† Department of Plant and Soil Sciences.‡ Department of Chemical Engineering.

426 Biotechnol. Prog. 2004, 20, 426−442

10.1021/bp034226s CCC: $27.50 © 2004 American Chemical Society and American Institute of Chemical EngineersPublished on Web 11/14/2003

signaling network was established via isolation of geneticmutants impaired in response to many different patho-gens (28-31). Most of this work used the model plantspecies Arabidopsis thaliana. Arabidopsis has numerousadvantages for research including a completely sequencedgenome, relatively facile transformation procedures, anda relatively short life cycle as compared with those ofmost plants (32). Genetic dissection of signaling identifiednumerous highly interconnected pathways with multiplelevels of feedback and complex regulation (33). Althoughit has been widely accepted that quantitative and dy-namic properties of this signaling network are important(34, 35), systems-level control of signaling has not previ-ously been addressed.

Dynamics, cross-talk, and systems-level emergent prop-erties have been studied in signaling networks throughuse of mathematical modeling (36-42). A standardapproach to modeling biological systems (43) is to identifymathematical functions that capture the relationshipsbetween important signaling components. These func-tions are used to construct a system of differentialequations that describe time-dependent changes in thesesignaling components. The resulting simulations arecompared with experimental data to validate the model.Validated models that accurately capture the biology canbe used to generate predictions of systems-level conse-quences of further experimental or genetic perturbations.

In this work, a model is developed of control over PCD.The particular experimental system chosen as the subjectof modeling was the Arabidopsis hypersensitive response(HR) to Pseudomonas syringae bacterial pathogens car-rying an avirulence (avr, disease resistance response-inducing) gene. In the HR, most cells in the leaf even-tually undergo PCD in response to inoculation of thewhole leaf with a high titer of avirulent pathogen. Thiswhole leaf response allows measurement of the extentof PCD in parallel with quantitation of associated signal-ing components. This experimental data was obtained ina companion study (44) from wild-type Arabidopsis andgenetic mutant plants altered in known feedback loopsinfected with isogenic bacterial strains varying only inthe avr gene. This rich data set was used to constrainmodel parameters by demanding consistency betweenexperimental and simulated results. The signaling cir-cuitry hypothesis underlying the modeling process ispresented first, followed by the experimental evidence forthis hypothesis. The process of model development is nextdescribed, followed by the details of the model. Simula-tions are subsequently presented and used to drawconclusions that led to refinements in understanding ofsignaling circuitry and guidance for further experimenta-tion. This improved understanding of PCD control willconstrain and inform future efforts in genetic engineeringof plants.

Current Working Hypothesis on SignalingCircuitry Used in Model Development

Figure 1 shows our current working hypothesis for thesignaling circuitry. The 11 signaling components picturedwere taken as variables in the mathematical model(Table 1). Three of these components were investigatedextensively in the companion study (44) that generatedthe dataset used for model validation (PCD, salicylic acid(SA), and H2O2). Qualitative information was used for afourth variable representing the level of the functionalprotein complex that recognizes the avirulence signalfrom the bacteria (avr‚R). The two biosynthetic precur-sors of SA, chorismate (CM) and the hypothetical alter-

native pathway precursor (Alt) that is likely phenylala-nine, were taken as having very large pools. There isexperimental support for the idea that SA biosynthesisis a very minor diversion of metabolic carbon flux fromthe shikimate and phenylpropanoid pathways (45). TF(triggering factor(s)) is a lumped variable for the poorlycharacterized signaling events linking avr‚R to PCD.Superoxide was included as a variable because of itsknown importance, even though it is very difficult toquantitate in a meaningful fashion in this system. Assuperoxide is a transient species, quantitation wouldneed to be in vivo. The active pool of superoxide isgenerated extracellularly (46). However, the chloroplastpool of superoxide is so large in comparison that in vivoquantitation of any other pool is impossible (Zhang andShapiro, unpublished data). Artificial reduction of thispool by growing plants in the dark would yield artifactsbecause the HR is light-dependent (3). Another modelvariable, apoplastic (extracellular) superoxide dismutase(SOD) activity, has not yet been measured in Arabidopsis.Chloroplastic and cytoplasmic SOD activities have beenmeasured from infected Arabidopsis leaves using in-gelassays (47). However, the identity of the apoplastic SODis not yet known. SOD activity measurements from

Figure 1. Signaling circuitry governing progression of theArabidopsis hypersensitive response to avirulent Pseudomonassyringae bacteria. Thin lines culminating in arrowheads rep-resent metabolic transformations or sequential signaling events.Dotted lines were used to indicate pathways thought to becomparatively minor in wild-type Arabidopsis. Thick lines wereused for regulatory circuitry. Lines terminating in a dash-headindicate negative regulation. Mutant blocks were indicated byslashes next to the name of the mutant. Abbreviations are asfollows: avr‚R ) functional form of complex genetically specifiedby an avr gene and its cognate R gene; TF ) lumped term fortriggering factor(s) that function downstream of avr‚R to triggerPCD; PCD ) programmed cell death; SOD ) apoplastic super-oxide dismutase activity; SA ) total salicylic acid; CM )chorismate; Alt ) hypothetical precursor in alternative SAbiosynthetic pathway (likely phenylalanine). The functions listedin Table 2 associated with each “arrow” are indicated. For clarityof presentation, most of the pathways for basal production anddegradation of model components are not pictured.

Table 1. Model Variables

name value at t ) 0

PCD 0.1SA 1O2

- 1H2O2 1SOD 1RNASOD 1RNAPCD 1avr‚R 0.01TF 0.01

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protein extracts of the apoplastic space might be possible.Model-based predictions of changes to SOD specificactivity (see Results and Discussion) will help guide theseexperiments.

The signaling pathway is initiated by inoculation withbacteria. Production of sufficient avr‚R to elicit the HRwas documented 75 min postinoculation (48). Accumula-tion of avr‚R leads to production of TF followed by PCD.PCD leads to superoxide production, likely by cellssurrounding the dying cells. Superoxide production leadsto SA biosynthesis in a manner that depends in part onthe NDR1 gene (arrow shown as interrupted by a slashlabeled “ndr1” to indicate blockage by a mutation in thisgene) and utilizes both pathways. Superoxide dismuta-tion produces H2O2, which is also assumed to upregulatethe alternative SA biosynthetic pathway. SA increasespotentiate TF action, thus constituting a positive feed-back loop (shown as thick line with arrowhead). Minorsignaling pathways pictured include PCD-independentproduction of very low levels of superoxide (in all cells)and SA-independent induction of apoplastic SOD activity.

Three potential negative regulatory circuits were con-sidered in this work, two of which are shown in Figure1. NPR1-dependent negative regulation of PCD is shownas an arrow interrupted by a slash labeled “npr1” toindicate that this pathway is disrupted in the mutant.The placement of this arrow was refined by results fromcomputational modeling (see Results and Discussion).The second negative regulatory circuit was included onthe basis of experimental evidence for a NDR1/LSD1/SA-dependent, NPR1-independent pathway. This pathwaywas taken to affect induction of SOD activity based onexperimental evidence discussed below. The third path-way was proven not to exist on the basis of results ofthis study and the companion experimental study (seeResults and Discussion) and was thus not included inFigure 1.

The last two variables in the model are putativetranscriptional events functioning in negative regulationdownstream of NPR1 (RNAPCD) and LSD1 (RNASOD),respectively. As LSD1 appears to be a transcription factorand NPR1 facilitates the binding of transcription factorsto DNA (49-51), assuming at least one gene inductionevent to be blocked downstream of mutations in each islogical.

Experimental Evidence for SignalingCircuitry Hypothesis

Programmed Cell Death. When a high titer ofavirulent bacteria is inoculated into a leaf, the leaf willeventually lose turgor, collapse, and wither. The leaf losesits water content because PCD of most cells in the leafresults in ion leakage that in turn causes water to leakto the extracellular spaces. This water is then lost viatranspiration. Leakage of cell contents through damagedplasma membranes is characteristic of plant PCD. Thisfeature is shared with mammalian oncotic PCD but isnot shared with apoptosis, the more common form ofmammalian PCD (52). This ion leakage is the basis ofthe most commonly used method of quantifying plantPCD (53-55). Samples are taken from infected leavesover a time course postinfection, washed to minimize ionleakage from cut edges, and placed in distilled water.Conductivity measurements of a series of samples typi-cally show a sigmoidal increase with time (44). Thesekinetics prove that PCD does not happen all at once.Instead, increasing numbers of cells die, culminating indeath of the whole leaf. An alternative, fluorescence-

microscopy-based assay of cell death based on internal-ization of a dye that crosses only compromised plasmamembranes confirmed that initial PCD events are indeeddispersed throughout the leaf (56). As such, at least threepopulations of leaf cells must be considered: the smallnumber of cells that die in direct response to pathogensignals, the surrounding cells that may produce oramplify second messenger signals upon perception ofPCD in neighboring cells, and the majority of cells in theleaf that will receive signals from either of the first twopopulations prior to death.

avr‚R Gene Interaction Specifies Molecular Rec-ognition of Pathogen. The signal from the bacteriumthat elicits plant PCD is the export of a bacterial avrprotein into the plant cell cytoplasm. Avr proteins aremade in the bacterial cytoplasm and subsequently trans-ported across both bacterial and plant plasma mem-branes and cell walls by a type III secretion system (57).Bacteria typically carry numerous avr genes (58). How-ever, plant resistance responses are elicited only if thehost plant carries a “Resistance” (R) gene that is thecognate match of the bacterial avr gene (59). In this work,the avr genes used were avrB and avrRpt2, which arerecognized by the R genes RPM1 and RPS2, respectively(59). Although avrB and avrRpt2 are delivered to theplant cytoplasm with similar kinetics (48), the HR ismuch faster when elicited by avrB (48, 60, 61). We havetaken advantage of these differences in the companionlaboratory studies used for model validation to triggerthe HR with either fast or slow kinetics by using isogenicbacterial strains that differ only in the avr gene. Themolecular basis for these differences has recently beenascribed to qualitative differences in protein-proteininteractions that involve additional components (62-64).

PCD Triggering Factor(s). The identity of the PCD“triggering factor(s)” that function downstream of the avr‚R gene-dependent signals is not certain. The best can-didate is calcium fluxes. Increases in cytoplasmic levelsof calcium were detected very soon after delivery of avrBto the plant cytoplasm; however, these fluxes were notdetectable within the first 3 h postinfection with bacteriacarrying avrRpt2 (48). It was speculated that a weaker,slower signal might not have been detected. Blockers ofcalcium channels have been shown to inhibit the HR (48,65), although the specificity of these reagents in plantshas been questioned (66). As the identity of signalingcomponents acting at this stage is uncertain, they havebeen “lumped” together into the variable triggeringfactor(s) (TF) in the model.

Reactive Oxygen Species and Superoxide Dis-mutase. Several other signaling components associatedwith the HR have been identified. Reactive oxygenspecies accumulate in plant leaves undergoing the HR.The first species to be produced is superoxide. Knockoutsin the biosynthetic enzymes for superoxide (NADPHoxidases) exhibit partial inhibition of the HR (67). Su-peroxide is produced extracellularly, and it rapidly dis-mutates to H2O2. This reaction is catalyzed by SOD (68).SOD leads not only to increases in dismutation rate butalso to increases in H2O2 yield because alternative redoxpathways compete effectively with uncatalyzed but notcatalyzed dismutation (69). SOD levels increase in re-sponse to infection (47).

In tissue culture cells responding to avirulent bacteria,a two-phase oxidative burst is seen prior to PCD (70).This burst was not sufficient to trigger PCD in culturedcells (71) and was not observed with whole plant studies.In infected leaves, reactive oxygen accumulation was seenin fluorescence microscopy-based assays only after the

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first PCD events (44, 60). Low-level production of reactiveoxygen was detected prior to this point using exquisitelysensitive electron-microscopy-based methods. However,abolishing this early production using a chemical inhibi-tor of NADPH oxidases did not affect the HR (48). Mostof the reactive oxygen produced during the HR appearsto be a consequence of PCD and is likely produced in cellssurrounding the dying cells (60). Nonetheless, very highlevels of reactive oxygen can induce PCD (46). Plantleaves likely accumulate these levels of reactive oxygenlate in HR progression. Genetic evidence (see below)indicates that high level reactive oxygen accumulationdoes indeed contribute to PCD at these time points.

Salicylic Acid. The major role of reactive oxygenaccumulation earlier in the HR appears to be in signalingleading to the production of the phytohormone SA. SA issynthesized from at least two pathways. In one pathway,the amino acid biosynthesis intermediate chorismate istransformed in two steps to SA (72). The first enzyme isisochorismate synthase (encoded by the Arabidopsis SID2gene). The second step was postulated by analogy withbacterial SA biosynthesis to involve an isochorismate:pyruvate lyase. The identity of the second pathway hasnot been established in Arabidopsis. However, in tobacco,SA is synthesized as an offshoot of the general phenyl-propanoid pathway (73, 74). Phenylalanine is convertedto trans-cinnamate in the first step of this pathway. In adiversion from the general phenylpropanoid pathway,trans-cinnamate is first conjugated (likely either toglucose or Coenzyme A). A chain shortening reactionthought to resemble fatty acid â-oxidation then producesconjugated benzoic acid. Benzoic acid is hydroxylated anddeconjugated to form SA. This pathway is the most likelycandidate to be the alternative SA biosynthetic pathwayin Arabidopsis.

Reactive oxygen production is known to elicit SAproduction in Arabidopsis in a manner that is dependentupon the NDR1 gene (60). We have recently establishedcapillary electrophoresis-based methods for quantifica-tion of SA from small numbers of infected Arabidopsisleaves. These methods were used in conjunction with sid2and ndr1 mutants to prove that NDR1 regulates boththe chorismate-derived and the alternative SA biosyn-thesis pathways (75). NDR1 encodes a putative trans-membrane protein (76) that is likely a signaling compo-nent acting downstream of reactive oxygen (60, 75).Reactive oxygen-independent, NDR1-independent SAbiosynthesis was also observed (75). We have also pos-tulated NDR1-independent activation of the alternativepathway for SA biosynthesis by H2O2 on the basis ofH2O2-dependent induction of SA biosynthesis in tobacco(77, 78). A small but significant NDR1-independentproduction of SA in Arabidopsis responding to treatmentsthat create multiple types of reactive oxygen in situprovided further evidence for this pathway (60).

SA Potentiation of TF Action Underlies a Posi-tive Regulatory Circuit. SA accumulation is requiredfor the HR to avrRpt2 but not for the HR to avrB whenassessed under standard conditions (5, 60, 79). Theweaker avr‚R signal apparently requires potentiation bySA in the case of avrRpt2 but not in the case of avrB. Apositive feedback pathway has indeed been documented(80, 81) by which SA potentiates the action of PCDtriggering factor(s) (60). We have, nonetheless, shownsubtle effects of potentiation with avrB in that PCD inndr1 mutant plants is delayed slightly at early timepoints postinfection (56). Potentiation was previouslyobserved in cultured cells and termed “agonist-dependent

gain control” with avirulent pathogen being the agonistand SA levels setting the gain of the response (82).

SA Accumulation Also Leads to Negative Regula-tion of PCD. Evidence for genetically specified negativefeedback came from analysis of ndr1 and npr1 mutantplants. Mutant ndr1 plants do not show a HR to bacteriacarrying avrRpt2 when assessed under standard condi-tions. However, the macroscopic leaf collapse associatedwith the HR to four other avr genes was greatly exag-gerated in extent of dryness and shriveling of the leaves(28, 33, 44). The hypothesis presented to explain theseresults was that a greater percentage of cells in the leafundergo PCD because the ndr1 mutation interruptednegative regulation of PCD. This hypothesis was subse-quently proven correct using the ion-leakage-based meth-ods for quantitative assessment of extent of PCD (44).An ion-leakage-based study using a single late time pointprovided preliminary evidence that an npr1 mutationalso interrupted negative regulation and a transgeneconferring overexpression of NPR1 enhanced negativeregulation (83). Time course studies using these methodswith wild-type, ndr1, npr1, and double mutant ndr1/npr1 plants showed that both mutations impaired nega-tive feedback (44). Impairment was to a greater degreewith ndr1 plants than with npr1 plants, and doublemutant plants resembled ndr1 plants. As the direct effectof the ndr1 mutation is to reduce SA production and npr1acts downstream of SA, these results proved that therewere at least two negative regulatory circuits.

Second SA-Dependent Negative Feedback Cir-cuit Operates via Changes in Level of ApoplasticSOD Activity. We have established that this secondnegative feedback loop operates via changes in the levelof SOD (44). Reduced induction of SOD in ndr1 mutantplants was inferred on the basis of experimentallyobserved reduction in H2O2 yield despite increases inPCD. Increased persistence of superoxide is likely theexplanation for enhanced cell death seen late in the HRin ndr1 mutant plants. A SA-dependent pathway forinduction of SOD had previously been discovered andshown to be blocked by mutations in the LSD1 gene (47).LSD1 encodes a putative transcription factor (84). Muta-tions in LSD1 lead to hair-trigger cell death in responseto superoxide (46). A paralog of LSD1 (LOL1) whoseprotein product binds to LSD1 has been identified. Thetwo related proteins are thought to act antagonisticallyas phenotypes of knockouts/overexpressors of LOL1 areopposite to those of LSD1 with regard to SOD accumula-tion and PCD (85). Placement of the lsd1 block insignaling circuitry is indicated in Figure 1. The assump-tion was made that LSD1-dependent induction of apo-plastic SOD will resemble that demonstrated for chloro-plastic and cytoplasmic SOD isoforms (47).

A different negative regulatory pathway was postu-lated by other researchers on the basis of increased SAseen in npr1 mutant plants relative to wild-type plants1-2 days postinfection with low levels of bacteria car-rying avrRpt2 (86, 87). Computational modeling (seeResults and Discussion) proved that the postulatedpathway for direct, NPR1-dependent negative autoregu-lation of SA biosynthesis could not exist under conditionsused to assess the HR. The experimental studies usedfor model validation also found no evidence for thispathway (44).

Possible Roles of Additional Signaling Compo-nents. Some signaling components suspected to affectthe HR were not modeled explicitly. Partial blockage ofthe HR in response to inhibitors of nitric oxide synthaseimplicated NO• in the HR (88, 89). We have shown that


NO• is produced extracellularly by cells that will die verysoon thereafter. Evidence was also presented that NO•

functions as a diffusible extracellular signal facilitatingcell-to-cell spread of the HR (56). As the effects of NO•

were subtle and the relevant direct targets were un-known, this qualitative data was not included in themodeling. As complete blockage of NO• production withNO• scavengers or biosynthetic inhibitors led to only a∼1 h delay in PCD (56), this omission was unlikely tohave had major effects on predictions of system dynamics.

Evidence for proteolytic events in HR control was alsonot considered explicitly. Genetic evidence of a role forubiquitin-mediated proteolysis in the HR and in diseaseresistance has been presented (31, 34, 90-94). However,it was argued (34) that proteolysis affects avr‚R ratherthan the HR per se on the basis of observed changes inaccumulation of RPM1. In support of this idea, the HRwas not completely compromised even in a null mutantin components required for proteolysis (31). As such, inthe modeling, these effects were subsumed in the variableavr‚R. Explicit treatment may be desirable in the futureonce further molecular details of this event are obtained.Caspase-like proteases were suggested to contribute toPCD on the basis of studies using caspase inhibitors (95,96) or measuring induction of caspase-like proteaseactivity (95). The experiments performed were not de-signed to distinguish whether the role for caspase-likeproteases was in early or late events. The experimentswith caspase inhibitors first measured PCD 8 h postin-fection. The experiments documenting virus induction ofcaspase-like protease activity used a temperature shiftprotocol with the viral infections known to lead to a largeburst of reactive oxygen. High levels of reactive oxygenare known to cause PCD in Arabidopsis in a fashion thatdepends on caspase-like protease activity (97). As it isnot clear at present whether caspase-like proteases are

necessary for hypersensitive cell death or are merely oneof many contributory causes to PCD very late in theresponse, they have not been included in the model.Modeling of additional signaling details will be necessaryas more is learned about this system. However, thedetails included at present captured the dynamics to theextent that good correspondence between simulated andexperimental data was achieved.

Model DevelopmentThe modeling process involved an iterative interplay

between biology and mathematics that is diagrammedschematically in Figure 2. Assumptions made on thebasis of experimental data presented in the precedingsections were recorded explicitly to facilitate periodicreevaluation. Variables were chosen as presented abovewith the goal of allowing the model to generate hypoth-eses that would be testable in the laboratory. Functionalforms were next chosen that captured relationshipsbetween signaling components. These functional formswere improved many times during the course of this workas new experimental data became available or as simula-tions proved functional forms to be inappropriate orassumptions to be wrong.

Initial parameter estimates were made on the basis ofavailable experimental data. A few estimates were madefrom literature data (e.g., the ratio between the rateconstants for catalyzed versus uncatalyzed superoxidedismutation (98)). However, most estimates depended onobservations of fold-increases seen in signaling compo-nents postinfection. As such, initial estimates wereconsidered to be in the correct order of magnitude butsubject to adjustment. When input data was purelyqualitative (e.g., TF being higher with avrB than withavrRpt2), parameter values were chosen to conform toknown qualitative relationships. As most of the available

Figure 2. Modeling strategy, showing dynamic interplay of computational and experimental approaches.


quantitative data was at the whole leaf rather than singlecell level, variables were normalized without consider-ation of cellular structure of the leaf. All reactions weremodeled with deterministic equations without probabi-listic or stochastic terms. Both the cellular structure ofthe leaf and probabilistic events will be considered infuture work.

Because the companion experimental studies wereperformed in parallel with this work, only a subset of thedata was available at the outset. As such, most parameterestimates were close to the values reported in Table 3prior to testing the model with the most recently obtaineddata. Therefore, adjustments made at this late stagefocused on only a few important parameters.

Two sorts of experimental data were useful for initialestimates of time delays. These estimates followed fromthe observation that each arrow on the diagram has an“input” and an “output”. If the “input” was not observedexperimentally prior to a given time point, then theprocess was taken to happen no earlier than that time.For many of the “arrows”, there was a direct consequenceon observable physiology that could be affected bymutants. The earliest time point at which a reproducibledifference in experimental measurements could be de-tected between mutant and wild-type plants was takenas the latest time before which the process must havebegun. The time difference between these two eventsmarked the maximum time delay for the process. Alldelays were taken as constant state delays (in continuoustime) set at a fixed number of hours. Delays of less than1 h were not modeled. In reality, delays between molec-ular level events are likely to be distributed around amean. When molecular events are better characterized,stochastic aspects can be introduced into the model ifnecessary to account for distributed delays.

To enable comparisons of the simulations to experi-ments with different basal levels of the model variables,the model variables were normalized such that the initial(basal) conditions for the variables were taken as 1 andchanges were calculated as fold changes above (or below)this level. The results of the simulations can be trans-formed linearly with respect to the measured levels ofthe model variables when necessary for comparisons toexperimental data. For PCD, the initial value was 0.1for ease of comparison to laboratory measurements thatwere already normalized. In the absence of precisemolecular definitions of avr‚R and TF, initial values of0.01 were taken for convenience.

Once the biological hypothesis had been rendered as aseries of delay differential equations, MATLAB code(available as Supporting Information at http://pubs.ac-s.org/) was written for numerical solution. Availablesolvers for delay differential equations (e.g., dde23) werevery slow in converging to a solution. This system ofdifferential equations is “stiff” (99), because it showsmajor fluctuations in levels of some of the variables onmuch shorter time scales than seen with other variables.The current MATLAB package does not include a delaydifferential equations solver for stiff systems (100). A newsolver (dde15s) was written specifically for this projectby Dr. Lawrence F. Shampine (Southern MethodistUniversity, Dallas, TX) based on ode15s. The code for thissolver is included in the Supporting Information to thispaper (http://pubs.acs.org). Stiff solvers for differentialequations break up the overall time interval into multiplepieces. The equations are solved one sequential piece ata time, using variable-length time steps. The MATLABcode runs approximately 150 times faster using dde15sas compared with dde23.

Simulations were subsequently performed of the timeevolution of all model variables. Response to the twodifferent avr genes was simulated by varying one pro-portionality constant (c81, see below). For comparisons toexperimental data, simulation of npr1 mutant plants orndr1 mutant plants was done by setting the parametersNPR1 or NDR1, respectively, to zero. These parameterscould take on different values in future studies ifcomparisons with experiments performed using allelicseries or transgenic lines with different levels of expres-sion of these genes were desired. Subsequently, theresponse of this system to changes in all parameters wassimulated. The approach was to vary each parameter(usually by 1 order of magnitude in each direction fromthe best available estimate) and simulate time evolutionof all states for each different level of each parameter.Initial analyses helped to diagnose errors in functionalforms or parameter estimates by identifying parametersfor which large changes had minimal effects on systemdynamics. Particularly interesting results from subse-quent analysis are presented (see Results and Discus-sion). The entire dataset is available as SupportingInformation (http://pubs.acs.org).

Model Description

The system of differential equations is shown in Figure3 with the details of functional forms used presented inTable 2. Table 1 lists the initial value of each variable(at time t ) 0, i.e., prior to inoculation with bacteria).The parameter values used in the simulations thatgenerated Figures 4 and 5 are listed in Table 3.

Equation 1. Programmed Cell Death. The unitchosen for PCD is related to the fraction of cells in a leafthat are dead at a given time point. The experimentalmeasurements in the companion study used an ion-leakage-based assay to measure cell death. As such,values for PCD were normalized to an initial value chosenfor ease of comparison to these measurements. HRprogression (dPCD/dt) was modeled separately for fourdifferent contexts (eq 1). These contexts were chosen onthe basis of the experimental data. For the earliest phaseof the response, in which experimental data showed noincrease in PCD, dPCD/dt was set to zero. A minimumlevel for TF (TFmin, see description of eqs 8 and 9 below)was set such that below this level, no PCD occurred.Although not explicitly treated in this paper, very highlevels of exogenous reactive oxygen can induce PCD inthe absence of bacteria. As late in the HR, superoxidedoes build up to sufficient levels to trigger PCD directly,another condition on this first rule was set as O2

- e

O2min- . A second rule was written to account for SA

potentiation of TF action being essential with avrRpt2but not avrB. This rule covered the condition of inter-mediate levels of TF (with superoxide levels still low tomoderate). We assumed that in potentiating TF action,SA would bind to a receptor. As such, kinetics were takento be Michaelis-Menten. However, experimental datashowed that SA levels began increasing up to severalhours prior to the first PCD events. We therefore allowedfor a delay between accumulation of SA and SA potentia-tion of TF action. In this way, the “effective concentra-tion” of TF (that considering SA potentiation) was forcedto exceed a dynamic threshold governing PCD triggering(f13). If exceeded, PCD commenced. For this situation, aflux representing the maximal rate of PCD in the absenceof negative regulation modifies a Michaelis-Menten-typefunction, allowing saturation at higher levels of TF (f11).This whole expression was multiplied by a term that


captured NPR1-dependent negative regulation of PCD(f12). This term is also Michaelis-Menten and allowssaturation at high levels of RNAPCD (the putative genefunctioning downstream of NPR1). The third rule ac-counted for PCD directly triggered by high level super-oxide accumulation. An additional term was added to theexpression in the second rule (f14). This term was chosento be saturable at high levels of superoxide, because thatwas the only functional form tried that allowed a matchto experimental data. Assuming direct proportionalitywith superoxide levels did not allow the solver to con-verge to a solution. Assuming that this effect wasproportional to the integral accumulation of superoxidewas also tried. The rationale was that superoxide mightoxidize or reduce a target that decayed very slowly, suchthat response was proportional to the time integral ofsuperoxide levels. However, making this assumptionresulted in this term having very little impact on systemdynamics. This result was inconsistent with experimentalobservations. Finally, a fourth rule was adopted to makecommitment of a cell to PCD independent of TF levelsabove a certain TF level. A TFmax was assumed in orderto satisfy this constraint.

Equation 2. Salicylic Acid. The time derivative ofSA levels incorporated terms for all known inputs to SAaccumulation. The choice was made to consider only thetotal pool of SA, including both free and conjugated forms.The majority of SA in an uninfected plant is known tobe in the form of a glucoside (101). However, thisglucoside is readily hydrolyzed in vivo to the activecompound (102). It is possible that glucosylation/deglu-cosylation will affect the active pool of SA. Although SAreadily diffuses across membranes, cellular or subcellularcompartmentation of SA biosynthesis/conjugation en-zymes or SA-binding proteins could also change theeffective level of SA. The diffusion across membranescomplicates experimental determination of compartmen-tal pools. As a result, no such determinations have beenreported. There is currently no evidence for differentdownstream cellular targets for the SA synthesized bythe two biosynthetic pathways. Because experimentalapproaches have not yet yielded sufficient data to provide

useful constraints relevant to these issues, they were notconsidered. SA was treated as existing in one pool. Thissimplification allowed comparisons with experimentalmeasurements of total SA levels. SA levels were normal-ized to the level in uninfected leaf tissue. This type ofnormalization was used for all state variables (exceptPCD, as discussed above, and avr‚R and TF, as discussedbelow) to facilitate comparisons with experimental data.

Equation 2 can be thought of as tripartite. The firstpart considers SA synthesized from the chorismate-derived pathway. An unregulated flux is written as aMichaelis-Menten-type term (f21). However, the assump-tion was made that pools of biosynthetic precursors ofSA are large and therefore not changing much duringthe simulations. As such, basal production was consid-ered constant, and its level was specified by the valuechosen for v21. This basal flux was multiplied by termsrepresenting regulatory influences experienced duringthe HR. We chose to multiply rather than add theseterms because of preliminary evidence that upregulationof the SID2 gene occurs as a consequence of bacterial (orfungal) infection (72). If it is upregulation of biosyntheticgenes rather than activation of novel pathways thataccounts for increased flux, multiplication is appropriate.Two Michaelis-Menten-type terms are included, one tocapture NDR1-dependent upregulation of SA biosynthe-sis in response to superoxide production (f22) and one tocapture NDR1 and reactive oxygen-independent regula-tion (f23). The alternative pathway was handled in ananalogous fashion, except that an extra Michaelis-Menten-type term was included to account for H2O2-dependent regulation (f27) as discussed above. A term forfirst-order decay (f28) was subtracted from the terms forbiosynthesis. Although we have recently proven that SAdegradation does indeed take place (75), as the pathway-(s) are unknown, first-order decay was assumed forsimplicity.

Equation 3. Superoxide. Time-dependent changesin superoxide levels were accounted for in eq 3. Althoughthis quantity cannot be measured experimentally in away that would be useful for modeling (see above), theimportance in the signaling network justified its inclu-

Figure 3. Model equations. Expressions used for each function are given in Table 2.


sion. Changes were considered after 1 h postinfection,as none were observed prior to that point in experimentalmeasurements. In vivo, reactive oxygen levels are acomplicated, dynamic balance between multiple path-ways for synthesis and multiple pathways for destruction(68). All basal synthesis was lumped into one flux (f31).Degradation of superoxide was treated with three sepa-

rate terms. Uncatalyzed dismutation was rendered asproportional to the square of superoxide concentration(f34) on the basis of in vitro studies (98). Dismutationcatalyzed by SOD (f35) was rendered using fractal kinetics(103). Exponents of 1.5 (rather than direct proportionalityseen in vitro) were used to account for restricted diffusionof SOD enzyme. As the relevant SOD is either in the cell

Table 2. Functional Forms

a For all i and j, vij represent fluxes, Kij represent equilibrium constants, kij represent kinetic rate constants, cij represent proportionalityconstants, and tij represent time delays. The values for i used in subscripts were the number of the equation in which a function orparameter first appeared. b TFmax replaces TF in f 11

/ .


wall or in the plasma membrane with the catalyticportion facing extracellularly, diffusion is highly re-stricted. This term accounts for pathogen-dependentincreases in dismutation via explicit dependence on SODlevels. The third term for decay is a lumped, first-orderterm that accounted for all alternative oxidants andreductants (f36). Parameter fitting reflected experimentaldata showing that this alternative decay was highlysignificant in plants impaired in pathogen-induction ofSOD (e.g., ndr1 mutant plants). Two other terms ac-counted for pathogen-dependent changes in superoxidelevels. The term taken to be large (on the basis ofexperimental evidence) captured superoxide produced asa consequence of PCD (f33). As no molecular details wereavailable to describe this step, direct proportionality todPCD/dt was assumed for simplicity. A second term wasincluded to account for PCD-independent upregulation

of superoxide production (f32). All superoxide accumula-tion prior to death of cells destined to undergo PCD wascaptured by this term. As this effect was known to bevery small, a Michaelis-Menten-type term was usedrather than direct proportionality, and parameters werefit appropriately.

Equation 4. Hydrogen Peroxide. Equation 4 cap-tured time-dependent changes in H2O2 levels. Terms forcatalyzed (f35) and uncatalyzed (f34) dismutation weretaken directly from eq 3, as H2O2 is the product of thesereactions. Although other reactions in the cell do produceH2O2, the physiologically relevant pool is that producedin the apoplast from superoxide. If infection does notchange these additional pools, they will contribute onlyto the background level seen in experimental data usedfor validation rather than to any changes seen on top ofthis background. H2O2 is destroyed by a diverse collection

Figure 4. Time evolution of major signaling components following infection with bacteria carrying avrB. Simulation of wild-typeplants (b) used parameter values given in Table 3. Simulation of npr1 mutant plants (3) and ndr1 mutant plants (9) had theparameters NPR1 or NDR1, respectively, set to zero. (A-D) Equations used for simulations were those in Figure 3. (E) AdditionalMichaelis-Menten-type terms were included to incorporate postulated direct negative autoregulation of SA biosynthesis. The inabilityof this plot to match experimental data was taken as evidence that this feedback loop does not exist, at least under the conditionssimulated. (F) The expression for SA-dependent, NPR1-dependent negative regulation of PCD was made to affect PCD resultingfrom high level accumulation of superoxide late in HR as well as PCD resulting directly from TF action. The inability of this plot tomatch experimental data was taken as evidence that this negative regulation affects only the latter.


of catalases and peroxidases. Which enzymes are present,how they change with infection, and how they arecompartmentalized were not known in sufficient detailto incorporate into the modeling. As such, we haveincluded only a lumped, first-order decay term (f41) forsimplicity.

Equation 5. Superoxide Dismutase. Time-depend-ent changes in SOD levels were modeled with eq 5. Abasal production of SOD enzyme (f51) and a first-orderdecay term (f54) were assumed. The SA-dependent SODinduction responsible for one of the negative feedbackcircuits (f52) was made to be a strong term by assumingdirect proportionality to the level of the postulated RNA.As the basal level of this RNA was set at 1 and basalflux was accounted for separately, a proportionalityconstant multiplies the expression (RNASOD - 1). The SA-independent induction of SOD was accounted for with aMichaelis-Menten-type term (f53).

Equations 6 and 7. Postulated Gene InductionEvents in Negative Regulatory Circuitry. Time-dependent changes in levels of the two postulated RNAs

were modeled in eqs 6 and 7. Basal rates of synthesis(f61 and f71) were assumed as well as first-order decay(f63 and f73). The terms that captured changes due toinfection (f62 and f72) were Michaelis-Menten. The up-stream genes for which mutants block induction (NPR1and LSD1) were included in the numerator, multipliedby proportionality constants, to account for wild-type(value ) 1), null mutants (value ) 0), and transgenic orallelic series in expression. Modeling presented hereinused only wild-type or null mutants. These terms weredesigned to saturate with increasing SA levels. Delaysin signaling were accounted for explicitly. As the basallevel of SA was set to 1, the SA-dependent terms reflectedincreases over basal levels. As basal fluxes were ac-counted for separately, this treatment was appropriate.

Equation 8 and 9. avr‚R and Triggering Factor-(s). The time evolution of avr‚R was modeled with eq 8.It is not clear what an “active” avr‚R-specified complexis or precisely when it is assembled. As such, thefunctional form assumed was a simple sigmoid multipliedby a proportionality constant (f81) with a term for first-

Figure 5. Time evolution of major signaling components following infection with bacteria carrying avrRpt2. Simulations performedand symbols chosen as in Figure 4.


order decay of the functional complex (f82). In eq 9, thetime derivative of TF was taken as the time derivativeof avr‚R incorporating a delay, multiplied by a propor-tionality constant (f91), minus a term for first-order decay(f92). As stated above, it is not known precisely what TFis; therefore, the direct proportionality between avr‚R andTF was assumed merely for simplicity. These parametervalues were all allowed to vary in order to fit simulatedto experimental data. The one constraint maintainedwith eqs 8 and 9 was that avrB elicited either higheramplitude (the option eventually chosen) or faster onset(conceivable) of TF levels than avrRpt2 did. Because ofthis choice, the plots of PCD versus time in Figures 4and 5 were scaled such that amplitudes resembledexperimental data. Amplitudes were doubled with avr-Rpt2 to account for the 2-fold difference in values for c81used with the different avr genes. As these steps insignaling are very active areas of investigation in numer-ous laboratories, further experimental details needed tocreate a mechanistic model are likely to be forthcoming.

These nine equations were adequate to capture theknown biology. The coding of these equations in MATLABallows easy incorporation of additional equations orchanges to terms or parameter values as simulations and/or experimental data necessitate these changes. Themodel is a tool for rigorous testing of consequences ofthese future revisions for consistency with all otheravailable data.

Results and DiscussionThe goals in constructing this model were to subject

specific hypotheses of signaling circuitry to rigoroustesting and formal analysis and to generate experimen-tally testable predictions. An important prerequisite forthese goals was that simulations using the model re-semble experimental data. These modeling efforts wereperformed in parallel with laboratory studies (44) mea-suring three of the most important components under theconditions taken for modeling. In these laboratory stud-ies, 2 × 107 mL-1 Pseudomonas syringae pv. tomatoDC3000 carrying either avrRpt2 or avrB on the identical

plasmid vector were used for infections. PCD (via ion-leakage-based methods), SA and H2O2 were measuredunder identical experimental conditions. Wild-type plantswere compared with npr1 and ndr1 mutant plants of thesame genetic background. Figures 4 and 5 (panels A-C)show simulations of precisely these experiments. Thecorrespondence with laboratory measurements was ex-cellent. The shapes of the time evolution curves for PCD,SA ,and H2O2, as well as the behavior of the variousmutants in response to the different bacterial strains,were well captured.

Cellular Compartmentation of Induced SA Bio-synthesis. The only exception to this close correspon-dence was highly informative in that it helped explaincellular compartmentation of induced SA biosynthesis.Predictions of SA accumulation data for the npr1 mutantdid not match the experimental data. In the simulations(panels 4B and 5B), SA levels were predicted to beidentical at all time points to those from wild-type plants.SA levels were indeed very close to those for wild-typeplants at most time points taken. However, experimentalmeasurements showed a marked impairment in SAaccumulation in npr1 plants at a few time points in thecompanion laboratory study. The difference was mostdramatic (5-fold) 4 h postinfection with bacteria carryingavrRpt2. The signaling circuitry diagram (Figure 1)provides no insight as to why these differences shouldhave occurred, since the npr1 mutant block in signalingis presented as being downstream of SA accumulation.

This comparison points to the limitations of one of theassumptions made. In the modeling, cellular compart-mentation of the leaf was ignored. Our interpretation ofthe experimental data (44) was that lifting of negativefeedback on PCD caused a population of cells to die tooearly to produce SA. As a result, SA levels were lowerprecisely at time points when npr1 plants would beexpected to have fewer cells alive and capable of makingSA. The model was not capable of coming to this result,because signaling was not modeled as a property ofindividual cells. Future modeling efforts will build uponresults presented herein via incorporation of a cellularautomata strategy (104) to account for the cellular natureof signaling.

Direct Negative Autoregulation of SA Biosynthe-sis Does Not Occur. Despite the one discrepancy notedabove, correspondence in all other aspects betweensimulated and experimental data was sufficiently goodto use the model as a tool to refine knowledge of signalingcircuitry. Evidence was discussed above for an additionalNPR1-dependent negative feedback loop for direct nega-tive autoregulation of SA biosynthesis. This hypothesiswas challenged by the lack of evidence for enhanced SAproduction in npr1 mutants in the experimental mea-surements made in the companion study (44). To test thishypothesis, model equations were modified by the addi-tion of Michaelis-Menten-type terms to eq 2 to accountfor these effects. With this additional term(s), no matterwhat values were chosen for parameters (aside from thetrivial result from setting parameters close to zero suchthat the terms had no effect), the simulated data couldnot match the experimental data. These disagreementscan be seen by comparison of panel C (equations preciselyas in Figure 3) with panel E (additional terms includedmodifying both SA biosynthesis pathways) in Figures 4and 5. In panels 4C and 5C, results with the npr1 mutantwere very similar to those with wild-type plants, as inthe experimental data. In panels 4E and 5E, the curvesare quite different. This in silico data supported ourcontention that this postulated feedback loop does not

Table 3. Values for Parameters

parameter value parameter value parameter value

v11 0.17 c25 1 K71 12K11 0.2 k21 0.25 c71 10K12 1 v31 0.3 k71 0.8K13 3 K31 0.25 t71 1K14 6 c31 0.05 c81

a 0.7/0.35c11 1 c32 6 k82 0.15c12 0.11 k32 0.000025 t81 2.5t11 1 k33 0.1 t82 1v21 0.35 k34 0.05 c91 1v22 0.27 k41 0.09 k92 0.15K21 0.01 v51 0.3 t91 1K22 3 K51 1 O2min 2.5K23 0.2 c51 0.3 TFmin 0.15K24 0.01 c52 0.1 TFmax 0.5K25 3 k51 0.3 NDR1b 1K26 0.2 v61 0.23 NPR1 1K27 3 K61 18.8 LSD1 1c21 8 c61 20 CM 1c22 1 k61 0.23 Alt 1c23 8 t61 3.5c24 1 v71 0.8

a This parameter was set to 0.7 for simulations of infection withbacteria carrying avrB and to 0.35 with avrRpt2. b NDR1, NPR1,and LSD1 were set to one in simulations of wild-type plants andto zero in simulations of the corresponding null mutants. Time isin units of hours. All other parameters are normalized relative toinitial values of variables listed in Table 1.


exist, at least under the conditions simulated. These datawere consistent with the experimental results that foundno evidence for the postulated feedback loop. As such,the corresponding terms were eliminated from the model.

Only One of Two Pathways Leading to PCD isSubject to SA-Dependent, NPR1-Dependent Nega-tive Feedback. Modeling also led to a second correctionto signaling circuitry hypotheses. Prior to the modelingefforts, it was not known which pathways leading to PCDwere regulated. Experimental evidence for timing of theregulatory effects had proven that the SA-dependent,NPR1-dependent negative feedback circuit must affectPCD resulting from TF action. However, it was not clearwhether PCD resulting from high level accumulation ofsuperoxide late in HR progression was also subject to thisnegative regulation. When the term incorporating theseeffects (f12) was taken to modify both pathways, thesimulations could not match the experimental data foravrB. Regardless of choices for parameters (aside fromtrivial results as discussed above), the relative magnitudeof curve amplitudes between ndr1 and npr1 plants wasthe opposite of that seen with experimental data (com-pare panel 4A, which does match experimental data, withpanel 4F, which does not). Correcting these two “arrows”on the circuitry diagram represents a major contributionto understanding of this signaling network and stronglyvalidates the systems biology approach taken herein. Inaddition, predictions of time profiles for SOD enzymeactivity changes will aid in identification of the relevantapoplastic SOD, and the “kinetic fingerprints” for RNAPCDand RNASOD induction will constrain the search for thepostulated genes.

Testable Predictions from Simulating Effects ofChanges to Individual Parameters. To generatefurther testable predictions, the effects of changes toindividual parameters were next assessed. The approachtaken was to vary each parameter stepwise, one at atime, and examine effects on all model variables. Param-eters were varied over 1 order of magnitude in eachdirection. For variations in one of the parameters (k21),this range prevented solutions from converging, so asmaller range was used (see Figure 7 caption). All datais available as Supporting Information (http://pubs.ac-s.org). Figures 6-9 show particularly interesting resultsof this analysis.

In Figure 6, the basal flux for SA biosynthesis fromchorismate (v21) was varied. Interestingly, the effectswere opposite with avrB (panel A) versus avrRpt2 (panelB). With either avr gene, raising the value up to 10-fold

had no discernible effects on the plot of PCD versus time.With avrB, reducing the value led to marked increasesin both midpoint and amplitude of the PCD versus timecurve. By contrast, with avrRpt2, reducing the valueabolished PCD. Figure 7 also predicted opposite effectson PCD with avrB (panel A) versus avrRpt2 (panel B).For this figure, the rate constant for SA degradation (k21)was varied over a 30-fold range (see Figure Caption forexplanation). Results with sid2 mutant plants (5) andnahG transgenic plants (that prevent SA accumulation)(79) were consistent with the simulated data in thatreducing SA accumulation did indeed have differentialeffects on the avrRpt2-elicited HR. These results high-light the ability of the model to compare and contrastsystems-level consequences of eliciting signaling withdifferent avr genes. As additional mechanistic details ofthe avr‚R gene interactions become available, thesecomparisons will offer insight into functional diversitywithin the plant R gene repertoire.

Figure 6. Changes in system dynamics due to variations inbasal flux of SA production from chorismate. The value for v21was varied 10-fold in either direction from the value listed inTable 3. (A) Response to bacteria carrying avrB. (B) Responseto bacteria carrying avrRpt2.

Figure 7. Changes in system dynamics due to variations inSA degradation. The value for the rate constant for first-orderdecay of SA (k21) was varied over a 30-fold range (10-fold downand 3-fold up from the value listed in Table 3). Higher valuesof k21 were not physically realistic in that if degradationexceeded production, no SA could ever accumulate. (A-C)Response to bacteria carrying avrB. (D-F) Response to bacteriacarrying avrRpt2.


In Figure 8, the strength of NDR1/superoxide-depend-ent upregulation of the chorismate-derived pathway forSA biosynthesis was varied via changes to the relevantproportionality constant (c21). As this constant directlymultiplies another constant that represents NDR1 levels,this figure can be considered simulation of a transgenicseries ranging from 10% to 1000% of wild-type NDR1activity. As expected, SA levels showed a strong depen-dence on the value chosen for c21 (panel B). This figureshows the simulated response to bacteria carrying avr-Rpt2. As expected with this avr gene, PCD was abolishedat the lowest values for c21 (panel A). The interesting andtestable result was that raising the value of c21 did notaffect PCD. The other panels of Figure 8 illustrate theprinciple of “kinetic fingerprints”. Very large differenceswere seen at some time points between simulationsmodeling NDR1 overexpressors, wild-type plants, andNDR1 underexpressors. Other time points showed neg-ligible differences. This type of data will be very helpfulin choosing time points for experimental tests of modelhypotheses. Large differences will be much easier todetect in the presence of experimental noise.

Figure 9 addresses a very important issue with regardto reengineering this pathway. The parameter that wasvaried (v11) was the maximal flux of cells undergoing PCDper unit time in the presence of maximal levels of TF,without consideration of negative regulation of PCD ordirect contributions to PCD from high superoxide levels.Varying this constant varies the extent of PCD (panelA), which leads to similar patterns of variation in theextent of superoxide (panel C) and H2O2 (panel D)accumulation. These results were expected. The insen-sitivity of SA levels to variations in v11 (panel B) wasunexpected. Manipulation of the HR without affectingSA levels would be very useful for engineering crop plantdisease resistance. SA increases are known to lead tonegative regulation of jasmonate biosynthesis and signal-ing, compromising defense against insects pests andcertain fungal pathogens (for review, see ref 33). Yet, SAis very important for defense against other pathogens.

However, this result is likely to be highly dependenton the functional form used to model superoxide-depend-ent induction of SA biosynthesis. At present, this rela-tionship is modeled with a Michaelis-Menten-type termthat approaches saturation under conditions used in thesimulations. As most superoxide production is a resultof PCD changes, changing this functional form willchange this result. If induction of SA synthesis wereinstead directly proportional to superoxide levels, muchstronger dependence on PCD would result. This cleardifference presents a potentially testable hypothesis.However, a gene affecting the maximal rate (as opposedto extent) of TF-dependent PCD would be required. Sucha gene has not yet been identified. Genetic screens formutants with altered PCD in many laboratories includingours will likely uncover such a gene. The model will at

Figure 8. Changes in system dynamics due to variations inefficacy of superoxide-dependent, NDR1-dependent upregulationof SA biosynthesis. The value for c21 was varied 10-fold in eitherdirection from the value listed in Table 3.

Figure 9. Changes in system dynamics due to variations inthe maximal rate of PCD. The value taken for the maximalnumber of cells undergoing PCD per unit time in the presenceof maximal levels of TF and without consideration of negativeregulation of PCD or direct contributions to PCD from highsuperoxide levels (v11) was varied 10-fold in either direction fromthe value listed in Table 3.


that point prove useful for predicting strategies formanipulating PCD while minimizing effects on nontargetpathways.

Conclusion

A differential-equations-based mathematical model ofHR progression and associated signal transduction wasdeveloped. The time evolution of PCD and major signal-ing components (H2O2 and SA) were captured faithfullyfor Arabidopsis responding to two different avirulentbacterial strains that differ markedly in initial strengthof HR triggering signal. Behavior of Arabidopsis mutantswas also captured faithfully, with the exception of SAaccumulation at a few well-defined time points using anpr1 mutant. This exception served to support currenthypotheses on cellular compartmentation of the signalingnetwork. Simulations helped to clarify negative regula-tory circuitry by supporting experimental evidence thatone postulated feedback loop specifying negative auto-regulation of SA biosynthesis does not exist. Anothernegative regulatory circuit was shown not to affect PCDcaused by high level accumulation of superoxide late inHR progression. The dynamic profiles of apoplastic SODactivity and two putative gene induction events werepredicted. Additional testable predictions were generatedby determining systems-level consequences of varyingeach parameter. These predictions will aid in design offurther experiments to test our knowledge of control ofthe HR. A detailed understanding of this signalingnetwork will ultimately be needed to underlie reengi-neering for improved disease resistance and to avoidinadvertent triggering of PCD in plant metabolic engi-neering projects.

Acknowledgment

The authors thank Dr. Lawrence F. Shampine (South-ern Methodist University, Dallas, TX) for writing andproviding the dde15s solver for stiff systems of delaydifferential equations. The authors thank Dr. BabatundeOgunnaike and Dr. Jeremy Edwards (University ofDelaware, Newark, DE) for useful discussions and criticalreading of the manuscript, respectively. This researchwas supported in part by a competitive grant to A.S. andP.D. from the Delaware Biotechnology Institute and inpart by a Young Professor Award from DuPont Corpora-tion to A.S. C.Z. was supported in part by a predoctoralresearch assistantship from the University of DelawareCollege of Agriculture and Natural Resources and by aUniversity Competitive Fellowship.

Supporting Information Available: Complete set of 3-Dplots from studies in which parameter values were variedsystematically. MATLAB code for model. MATLAB code for thedde15s solver for stiff systems of delay differential equations(courtesy of Dr. Lawrence F. Shampine). This material isavailable free of charge via the Internet at http://pubs.acs.org.

Accepted for publication October 9, 2003.

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