Journal of Theoretical Biology 273 (2011) 188–196

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Journal of Theoretical Biology

0022-51

doi:10.1

� Corr

E-m

journal homepage: www.elsevier.com/locate/yjtbi

Transient pulse formation in jasmonate signaling pathway

Subhasis Banerjee, Indrani Bose �

Department of Physics, Bose Institute, 93/1, A. P. C. Road, Kolkata 700009, India

a r t i c l e i n f o

Article history:

Received 16 March 2010

Received in revised form

23 December 2010

Accepted 23 December 2010Available online 29 December 2010

Keywords:

Feedback loop

Transient pulse

Stochastic dynamics

Defense response

Signaling pathway

93/$ - see front matter & 2010 Elsevier Ltd. A

016/j.jtbi.2010.12.037

esponding author. Tel.: +91 033 2303 1184;

ail address: indrani@bosemain.boseinst.ac.in (

a b s t r a c t

The jasmonate (JA) signaling pathway in plants is activated as defense response to a number of stresses

like attacks by pests or pathogens and wounding by animals. Some recent experiments provide significant

new knowledge on the molecular detail and connectivity of the pathway. The pathway has two major

components in the form of feedback loops, one negative and the other positive. We construct a minimal

mathematical model, incorporating the feedback loops, to study the dynamics of the JA signaling

pathway. The model exhibits transient gene expression activity in the form of JA pulses in agreement with

experimental observations. The dependence of the pulse amplitude, duration and peak time on the key

parameters of the model is determined computationally. The deterministic and stochastic aspects of the

pathway dynamics are investigated using both the full mathematical model and a reduced version of it.

We also compare the mechanism of pulse formation with the known mechanisms of pulse generation in

some bacterial and viral systems.

& 2010 Elsevier Ltd. All rights reserved.

1. Introduction

Plants are frequently subjected to a number of stresses likeattacks by pests or pathogens and wounding by animals whichcause damage to the plant cells. In higher plants, the response todamages is mediated via the jasmonate (JA) signaling pathway(Browse, 2009; Devoto and Turner, 2003; Lorenzo and Solano,2005; Farmer et al., 2003; Turner et al., 2002). On activation of thepathway, there is a rapid but transient accumulation of JAs in theinfected/wounded cells. This leads to the expression of JA-respon-sive genes involved in defense-related processes. More generally,JAs are produced in response to a number of biotic and abioticstresses and also play active roles in regulating developmentalprocesses. The JA signaling pathway is an integral component of aplant’s defense system but till recently there were large gaps in ourunderstanding of how the pathway operates. Two experimentalstudies (Chini et al., 2007; Thines et al., 2007), carried out on themodel plant Arabidopsis thaliana, now provide significant knowl-edge on the molecular details of the signaling pathway which helpsin the elucidation of its functional features.

Fig. 1 shows a schematic diagram of the core components of theJA signaling pathway (Farmer, 2007). JA responses typically involvechanges in the expressions of hundreds of genes (Browse, 2009).Transcription factors (TFs) like MYC2 regulate the expressions ofJA-responsive genes including those promoting JA biosynthesis.Under normal circumstances, MYC2 is non-functional due to

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fax: +91 033 2350 6790.

I. Bose).

binding by the JASMONATE-ZIM-domain (JAZ) repressor proteinsand the JA levels in cells are low. On wounding, damage-inducedsignals initiate JA biosynthesis so that JA levels are elevated in cells.This results in the formation of the jasmonate-L-isoleucine (JA-Ile)complex which mediates the binding of CORONATINE INSENSITIVE1(COI1) and JAZ proteins. COI1 forms part of an enzyme complexSCFCOI1 which tags the bound JAZ proteins with ubiquitin markingthem for destruction. The degradation of JAZ proteins liberates theTF MYC2 and possibly other TFs involved in JA-induced geneexpression. Thus, enhanced JA levels result in the activation ofthe JA signaling pathway through the destruction of the JAZproteins, the repressors of the pathway. There are two primaryregulatory loops a and b in the JA signaling pathway (Fig. 1). Loop bis the newly discovered (Chini et al., 2007; Thines et al., 2007;Farmer, 2007) negative feedback loop involving MYC2 and JAZproteins. MYC2 activates the synthesis of the JAZ proteins whereasJAZ binds MYC2 and inhibits its activity as a TF. The synthesis of JA isautoactivating via the positive feedback loop a since a JA signalgives rise to free MYC2 which in turn activates the genes respon-sible for JA biosynthesis. The combination of positive and negativefeedback loops a and b gives rise to transient gene expressionactivity so that JA accumulation is in the form of a pulse.

In this paper, we develop a minimal mathematical modelincorporating the core components of the JA signaling pathway(Farmer, 2007) shown in Fig. 1. We specially investigate how theregulatory loops affect the gene expression dynamics. Based on ourresults, we make testable predictions on the dependence of theamplitude and duration of the transient JA pulses on the keyparameters of the model. There are other examples of transientpulse formation in gene regulatory networks which include the

JA−Ile etc.

JA

Defence response

JAZ

a

bTF

COI1 Protein

Fig. 1. Core components of the JA signaling pathway (Farmer, 2007). Transcription

factor (TF), MYC2, activates the synthesis of both JA and JAZ. The arrow sign denotes

activation and the hammerhead sign repression. The function of the components is

described in the text.

S. Banerjee, I. Bose / Journal of Theoretical Biology 273 (2011) 188–196 189

development of competence in the bacterial population Bacillus

subtilis (Smits et al., 2005; Suel et al., 2006; Maamar et al., 2007),infection of a host cell by the HIV-1 virus with a subsequent choicebetween two distinct cell fates, latency and lysis (Weinberger,2007; Weinberger et al., 2005, 2008), and regulation of theSalmonella pathogenecity by the PhoP/PhoQ two-componentsystem (Shin et al., 2006). The transient pulse in each case is thatof a key regulatory protein: ComK (competence development), Tat(HIV-1 virus) and phosphorylated PhoP (Salmonella pathogene-city). The regulatory proteins reach sufficiently high concentrationsin the peak regions of the pulses to activate competence develop-ment in B. subtilis, lysis in the host cell infected by the HIV-1 virusand virulence in the case of salmonella pathogens. In a B.subtilis

population subjected to stress, competence develops in a fraction ofcells involving the uptake of DNA from the environment and itsintegration into the bacterial genome. This confers new traits onthe bacteria which helps in bacterial survival under stress. Acombination of mathematical modeling and experimentalapproaches provides new insight on the mechanisms of pulseformation in the cases considered. The mechanisms differ in detailbut there is one common feature, namely, the presence of a positivefeedback loop involving the key regulatory protein.

The transient nature of a regulatory pulse is determined by twoopposing influences. In the case of B. subtilis, the competence eventsare generated by two feedback loops, one positive and the othernegative. Positive feedback amplifies the ComK level whereasnegative feedback brings it down to that of the vegetative (non-competent) state. The HIV circuit has been proposed to functionlike a feedback resistor (Weinberger, 2007) to explain transient Tatactivity. Forward reactions (e.g., acetylation of Tat) constituting thepositive feedback loop amplify the Tat level whereas stronger backreactions (deacetylation, degradation and unbinding of the reg-ulatory molecules) eventually deactivate the circuit. The two-component PhoP/PhoQ system controlling Salmonella pathogeni-city operates on similar principles (Shin et al., 2006). These knownmechanisms of transient pulse formation provide the motivationfor identifying the mechanism underlying transient pulse genera-tion in the JA signaling pathway. In Section 2, we develop a minimalmathematical model of the JA signaling pathway to study how JApulses are formed. The dependence of the pulse characteristics likeamplitude, duration and peak time on the key parameters of themodel is computed. We further consider a reduced version of the

general model to capture the essential features of the dynamics ofpulse formation. In Section 3, we investigate the dynamics of the JAsignaling pathway using stochastic approaches. Section 4 containsconcluding remarks and a comparison of the mechanisms of pulseformation in different signaling pathways.

2. Mathematical model of JA-signaling pathway

In the deterministic approach, the dynamics of the JA-signalingpathway are determined by a set of differential equations describ-ing the rates of change in the concentrations of the key molecularcomponents in the pathway (Fig. 1) with respect to time. Thebiochemical reactions to be considered are

JZþM "k1

k2

M�JZ ð1Þ

JAþ IL"k3

k4

JAIL ð2Þ

JZþ JAIL"k5

k6

JZ�JAIL-k7

JAIL ð3Þ

JZ, JA and M denote the molecular species JAZ, JA and MYC2. IL,M�JZ, JAIL and JZ�JAIL represent isoleucine, the bound complex ofJAZ and MYC2, the JA-Ile complex and the complex of JAZ with JA-Ile respectively. The same symbols represent the concentrations ofthe respective molecular species. The rate constants k1–k7 areassociated with the different forward and backward reactions. Twoother reactions which we take into account describe the degrada-tion of JAZ and JA. As mentioned in the Introduction, we develop aminimal model of the JA-signaling pathway to probe the mechan-ism of JA pulse formation. In the following, we point out thesimplifications made in developing the model and the justificationsthereof. JA and JA-Ile levels are known to increase dramatically(� 25�fold increase within 5 mins of tissue injury) in response toboth mechanical wounding and herbivory (Koo and Howe, 2009).The remarkable speed of response suggests that all the biosyntheticenzymes required for the production of JA/JA-Ile are available in theresting cells. In our minimal model, the enzymatic activity isimplicit in the effective reactions considered with the enzymeconcentrations absorbed in the appropriate rate constants. Theconjugation of JA to Ile is mediated by the enzyme JASMONATERESISTANT1 (JAR1) (Staswick, 2008) which is described by theeffective binding reaction in Eq. (2). Recent reports (Browse, 2009;Chini et al., 2007; Thines et al., 2007; Farmer, 2007; Chini et al.,2009) have conclusively established that JA-Ile directly inducesthe binding of the JAZ proteins with COI1 which results in thedestruction of the repressor proteins. These processes are com-bined in the effective reactions contained in Eq. (3). Experimentalevidence of the direct binding between COI1 and JA-Ile has not beenreported so far (Chini et al., 2009) though a molecular mechanismfor the formation of the bound complex of JA-Ile, COI1 and JAZ hasbeen proposed recently (Chung et al., 2009; Yan et al., 2009). TheJAZ proteins are distinguished by the presence of a highly con-served Jas domain which mediates protein–protein interactionswith both TFs like MYC2 and COI1 (Chung et al., 2009). It appearsthat the COI1 and MYC2 proteins compete for interaction with theJas motif (Chini et al., 2009). The presence of JA determines theoutcome of the competition since the COI1-Jas domain interactionrequires the presence of JA-Ile whereas MYC2 interacts in ahormone-independent manner.

Apart from the reactions (1)–(3), the model also takes intoaccount the synthesis of JAZ proteins through the TF (MYC2)regulated expression of the JAZ genes and the biosynthesis of JA.We do not model the full octadecanoid pathway for JA biosynthesis

Fig. 2. Transient pulses of JA (black) and JZ (red) versus time in hours. JA and JZ have

the units of concentration. The inset of the figure shows the mean values of JA and JZ

over time with averages taken over 10 individual time traces. (For interpretation of

the references to color in this figure legend, the reader is referred to the web version

of this article.)

Fig. 3. Variation of JA versus time in hours for values of b2 ¼ 10, 20, 30 and 40. The

peak height of the pulse increases for increasing values of b2. JA has the unit of

concentration.

S. Banerjee, I. Bose / Journal of Theoretical Biology 273 (2011) 188–196190

which is activated by damage-induced signals (Browse, 2009;Wasternack, 2007). One major feature of the pathway is that theJA biosynthesis genes are regulated by TFs like MYC2, thusconstituting a positive feedback loop. In recent studies (Pauwelsand Goossens, 2008) MYC2 has been identified as a transcriptionalactivator of the JA biosynthesis gene LOX3. This finding matchesearlier observations (Lorenzo et al., 2004) that JA-induced LOX3expression is reduced in the MYC2-defective mutant jin1. Thedifferential rate equations of the model are given by

dðJZÞ

dt¼

b1M

Km1þMþk2M�JZ�k1JZ:Mþk6JZ�JAIL�k5JZ:JAIL�g1JZ

ð4Þ

dðJAÞ

dt¼

b2M

Km2þMþk4JAIL�k3JA:IL�g2JA ð5Þ

dðM�JZÞ

dt¼ k1M:JZ�k2M�JZ ð6Þ

dðJAILÞ

dt¼ k3JA:IL�k4JAILþðk6þk7ÞJZ�JAIL�k5JAIL:JZ ð7Þ

dðJZ�JAILÞ

dt¼ k5JZ:JAIL�ðk6þk7ÞJZ�JAIL ð8Þ

The first term in Eq. (4) describes JAZ synthesis due to MYC2-regulated gene expression. The TF binding/unbinding events occuron a time scale much faster than that of protein synthesis anddegradation. One can thus assume that steady state conditionsprevail in the formation of the TF-gene complex. The MYC2-mediated synthesis of JA is represented as an effective processby the first term in Eq. (5). The justification of the first terms in Eqs.(4) and (5) lies in the experimental observations (Thines et al.,2007; Farmer, 2007; Chini et al., 2007; Pauwels and Goossens,2008) that MYC2 activates the expression of JAZ genes as well as JAbiosynthesis genes like LOX3. The mathematical forms representsteady state concentrations of the bound complex of the appro-priate gene with MYC2 acting as a TF. The rate constants b1 and b2

denote the maximum rates of JAZ and JA synthesis. The synthesisrate is half-maximal when Km1, Km2¼M, the concentration of theregulating protein MYC2. There are two conservation equations:

Mt ¼MþM�JZ ð9Þ

ILt ¼ Ilþ JAILþ JZ�JAIL ð10Þ

where Mt and ILt are the total concentrations of MYC2 andisoleucine respectively in free as well as bound forms. The rateconstants g1 and g2 in Eqs. (4) and (5) are the degradation rateconstants for JAZ and JA.

We now investigate whether the mathematical model, describedin terms of Eqs. (1)–(10), can reproduce transient pulse formationconsistent with experimental observations in wounded plants (Kooand Howe, 2009; Chung et al., 2008; Glauser et al., 2008). We find onsolving the equations that JA accumulation in the form of a pulse ispossible. Fig. 2 shows one such transient pulse for the parametervalues b1 ¼ 20, b2 ¼ 30, k1¼0.2, k2¼0.001, k3¼0.1, k4¼0.1, k5¼0.1,k6¼1.0, k7¼1.0, Km1¼1, Km2¼1, g1 ¼ 0:1, g2 ¼ 0:4 in appropriateunits. In the absence of experimental estimates of the different rateconstants, we have varied the parameter values over extended rangesto show that pulse formation is quite general. The concentration of JAin the pulse initially rises with time, reaches a maximum value andthen decays to reach a low steady state level. Fig. 2 also shows a plot ofJZ values versus time. The transient nature of the JA pulse is anoutcome of competing negative and positive feedback loops. Thepositive feedback loop (a in Fig. 1) in the JA signaling pathway has theeffect of amplifying the JA level whereas the negative feedback loop (bin Fig. 1) brings back the level to a stable low value. An individualpulse is distinguished by quantities like amplitude (maximal value),

temporal duration and peak time, tm i.e., the time at which themaximal value of the variable being measured is achieved. Theduration time tD is measured as tD¼tmax2�tmax1, where tmax1 andtmax2 are respectively the time points at which the rates of increaseand decrease attain their largest values. Since positive feedback hasthe effect of amplifying JA levels/activity, the amplitude of a pulse canbe increased by strengthening the positive feedback. Fig. 3 shows howthe JA concentration changes as a function of time for different valuesof b2, the maximal rate of JA synthesis. The rest of the parameters hasthe same values as in the case of Fig. 2. Fig. 2 is obtained by solving thedifferential equations (4)–(8) with the initial (t ¼ 0) conditions JA¼2,M¼100 and Il¼20. The values assumed for Mt and ILt are Mt¼200 andILt¼20. An initial dose of JA molecules is essential to free MYC2 so thatactivation of the JA signaling pathway is possible. Wounding signalsstimulate the biosynthesis of JA which provides the initial JA input.

S. Banerjee, I. Bose / Journal of Theoretical Biology 273 (2011) 188–196 191

Once free MYC2 is available, JA biosynthesis is further stimulated viathe positive feedback loop shown in Fig. 1. Our model calculationsstart with the state which already has free MYC2 and we focus on howthe activation and deactivation of the JA pulse occurs due to thedynamics of the coupled positive and negative feedback loops. If theinitial JA and MYC2 values are increased, the amplitude of thetransient JA pulse increases. As b2 becomes larger, there is a greateraccumulation of JA resulting in an increase in the amplitude of the JApulse. The magnitude ofb2 is a measure of the strength of the positivefeedback. The pulse amplitude is reduced considerably withdecreased values of b2. For b2 ¼ 0 (no positive feedback), there isno pulse formation.

Fig. 4 shows the variation of JA concentration versus time fordifferent values of b1, the maximal rate of JAZ synthesis. The otherparameter values are the same as in the case of Fig. 2. Withincreasing values of b1, the amount of JAZ proteins goes up andthere is more repression of MYC2 activity. This leads to a reductionin the duration of the JA pulse. The change in the amplitude is,however, not as prominent as that when b2 is varied. This is moreevident in Figs. 5(a) and (b) in which the amplitude of the JA pulse isplotted againstb1 andb2, respectively keeping the other parametervalues the same as in the case of Fig. 2. Figs. 5(c) and (d) exhibit thevariation of the peak time tm versus b1 and b2 respectively. Themagnitude of tm decreases considerably as b1 is increased. Thedependence of tm on b2 is, however, insignificant. Figs. 5(e) and(f) show a plot of tD, the duration of the JA pulse, versus b1 and b2.The dependence of tD on b2 is negligible. Fig. 6 shows the variationof JA concentration versus time as the degradation rate constant g2

is changed. Similar figures may be obtained by varying the otherparameters of the model. For the set of parameter values con-sidered, the most prominent changes are obtained by varying themaximum synthesis rates b1 and b2. Figs. 3–6 have been obtainedwith the total concentrations fixed at the values Mt¼200, andILt¼20 in appropriate units. Table 1 lists all the parameters of thedynamical model, their description as well as units and the rangesof values of the parameters for which transient pulse formation hasbeen checked. The different parameter units are expressed in termsof concentration ([c]) and time ([t]) units. There are presently nomethods available to record the cellular concentrations of JA, JA-Ileetc., the quantities measured are amounts per gram fresh weight. Inour model study, the JA amounts have the units of concentration.

To analyze the features of the model better, we reduce it to atwo-variable model described by rate equations which are in

Fig. 4. Variation of JA versus time for different values of b1 ¼ 5, 10, 20, 30 and 40.

dimensionless form.The reduction rests on the fact that themolecular complexes M�JZ, JAIL and JZ�JAIL attain their steadystate concentrations on timescales which are comparably shorterthan those over which the JA and JAZ reach their steady stateconcentrations. Therefore, one can take the time derivatives in Eqs.(6)–(8) to be zero so that the complexes M�JZ, JAIL and JZ�JAIL,attain their steady state concentrations. The reduced model isdescribed by the rate equations

dðJZÞ

dt¼

b1M

Km1þM�dJZ

JA:JZ

JAþK1þJZ

G:JA

�g1JZ ð11Þ

dðJAÞ

dt¼

b2M

Km2þM�g2JA ð12Þ

with G¼ ðk6þk7Þ=k5, K1¼ k4=k3 and dJZ ¼ k7ILt=G. Also,Mt ¼Mð1þðk1=k2ÞJZÞ and ILt ¼ ILð1þðk3=k4ÞJAð1þ JZ=GÞÞ. Rescalingtime with dJZ and the concentrations JZ and JA withG, we obtain thedimensionless rate equations:

dðJZÞ

dt¼

b1

kn1ð1þK uJZÞþ1�

JA:JZ

JAþK2þ JA:JZ�d1JZ ð13Þ

dðJAÞ

dt¼

b2

kn2ð1þK uJZÞ�d2JA ð14Þ

where t, JZ and JA are now dimensionless variables with thedimensionless parameters shown in Table 2.

Eqs. (13) and (14) can be further simplified to

dðJZÞ

dt¼

a1

1þm1JZ�

JA:JZ

JAþK2þ JA:JZ�d1JZ ð15Þ

dðJAÞ

dt¼

a2

1þm2JZ�d2JA ð16Þ

The parameters a2, a2, m1 and m2 are defined in Table 2. We firstconsider the case d1 ¼ 0 as degradation of the JAZ proteins mostlyoccurs via the second term in Eq. (13) describing ubiquitin-mediated degradation on the binding of the SCFCOI1 complex toJAZ proteins. The steady state of the dynamical system is obtainedby putting the temporal rates of change, dðJZÞ=dt and dðJAÞ=dt, to bezero. There is only one physical steady state solution given by

ðJZÞs ¼Aþ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiA2þ4a1a3m1ða3þK2Þ

p2a3m1

, A¼ a1a3þa1K2m2�a3 ð17Þ

ðJAÞs ¼a3

1þm2ðJZÞs, a3 ¼

a2

d2ð18Þ

The local stability of the steady state can be established usingthe standard procedure of linear stability analysis (Strogatz, 1994).The checking of global stability is a more difficult problem as onehas to demonstrate that the steady state is approached from allinitial conditions. For d1 ¼ 0, there is only one physical steady stateto which the phase space trajectories starting from all physicallyrelevant initial states are expected to converge. We have used thesoftware package XPPAUT [http://www.math.pitt.edu/�bard/xpp/xpp.html] to numerically check the stability of the steady statefor several sets of parameter values. When d1a0 in Eq. (15), againfor several sets of parameter values there is only one physicalðJA40,JZ40Þ stable steady state solution. Table 2 shows theranges of values of the different parameters for which the stabilityproperties have been checked numerically. Fig. 7 shows the phaseportrait obtained from Eqs. (15) and (16) with the parameter valuesa1 ¼ 2:5, a3 ¼ 50, m1 ¼ 50, m2 ¼ 10, K2¼1000, and d1 ¼ 0:015. Thenullclines, obtained by putting dðJAÞ=dt¼ 0 and dðJZÞ=dt¼ 0, inter-sect at a single point, the so-called fixed point of the dynamics. Twotrajectories (shown by dotted lines) with two different initialconditions converge to the fixed point in the course of time. Each

Fig. 5. Amplitude of JA pulse versus (a) b1 and (b) b2; peak time tm in hours of the JA pulse versus (c) b1 and (d) b2; duration tD in hours of JA pulse versus (e) b1 and (f) b2.

S. Banerjee, I. Bose / Journal of Theoretical Biology 273 (2011) 188–196192

trajectory depicts the values of JA and JZ at different time pointsobtained by solving Eqs. (15) and (16). Fig. 8 shows the magnitudeof JA versus time for the same parameter values. The transient pulsereaches a peak value due to an initial surge and the system finallysettles down into a stable steady state with the value of JA muchlower than the peak value.

3. Stochastic dynamics

The single steady state characterized by a low value of JA is stableunder small perturbations. Under sufficiently strong perturbations,

a large-amplitude JA pulse is excited with eventual return to thestable steady state. The random nature of the biochemical eventsinvolved in gene expression gives rise to fluctuations in proteinlevels (Kaern et al., 2005; Raj and van Oudenaarden, 2008) which actas perturbations. The simplest way to take the effect of fluctuationsinto account is to incorporate an additive noise term in either of thedynamical equations (15) and (16). The noise term further incor-porates the effects of transient perturbations originating in wound-ing signals. The most prominent effect is obtained with the noiseterm added to Eq. (15). A possible reason for this is that JAZrepresses the synthesis of both JA and JAZ so that fluctuations in theJAZ levels affect both the JA and JAZ levels. The stochastic dynamics

Fig. 6. Variation of JA versus time in hours for different values g2 ¼ 0:1, 0.2, 0.3 and

0.4 of the degradation rate constant. The amplitude of the JA pulse decreases for

increasing value of g2. JA has the unit of concentration.

Table 1Parameters, their meanings and defining formulae (the units are given in brackets,

[c]¼concentration, [t]¼time).

Parameter Range of values Meaning/defining formula

k1 ,k2

1

½c�½t�,1

½t�

� � 0.01–0.2, 0.001–

0.2

Rate constants for the formation and

dissociation respectively of the bound

complex (M_JZ) of JAZ and MYC2 proteins

(defined in Eq. (1))

k3 ,k4

1

½c�½t�,1

½t�

� � 0.1–0.8 Rate constants for the formation and

dissociation of the bound complex (JAIL) of

JA with IL (defined in Eq. (20))

k5 ,k6

1

½c�½t�,1

½t�

� � 0.1–0.8,0.1–1.0 Rate constants for the formation and

dissociation respectively of the complex

JZ�JAIL of JAZ proteins with JAIL (defined in

Eq. (3))

k7

1

½t�

� � 0.1–1.0 Rate constant for ubiquitin-mediated

degradation of JAZ proteins (defined in

Eq. (3))

g1 ,g2

1

½t�,1

½t�

� � 0.1–1.0 Degradation rate constants of JAZ proteins

and JA respectively (defined in Eqs. (4) and

(5))

b1 ,b2

½c�

½t�,½c�

½t�

� � 5–100 Maximum rates of JAZ and JA synthesis

(defined in Eqs. (4) and (5))

Km1 ,Km2

ð½c�,½c�Þ

0.1–1.0 Synthesis rate of JAZ and JA is half-maximal

when Km1, Km2¼M, the concentrations of

the regulatory protein MYC2

Mt ,ILt

ð½c�,½c�Þ

– Total concentrations of MYC2 and

isoleucine respectively which are defined in

Eqs. (9) and (10)

Table 2Rescaled parameters and their definitions.

Parameter Range of values Definition Abbreviation

b1 b1

dJZGdJZ ¼

k7ILt

Gb2 b2

dJZGG¼

k6þk7

k5

kn1 Km1

MtMt ¼M 1þ

k1

k2JZ

� �

kn2 Km2

MtILt ¼ IL 1þ

k3

k4JA 1þ

JZ

G

� �� �

K uG

k1

k2

K2 5–2000 K1

Gd1 0–0.5 g1

dJZ

d2 1–10 g2

dJZ

a1 1–200 b1

kn1þ1a2 1–200 b2

kn2þ1m1 1–1000 kn1K u

ðkn1þ1Þm2 1–1000 kn2K u

ðkn2þ1Þ

Fig. 7. Phase portrait defined by Eqs. (15) and (16). The solid lines represent the

nullclines (dðJAÞ=dt¼ 0 ðredÞ, dðJZÞ=dt¼ 0 ðblackÞ) intersecting at one fixed point.

The fixed point describes a stable steady state. Two typical trajectories (dotted lines)

are shown with arrow directions denoting increasing time. (For interpretation of the

references to color in this figure legend, the reader is referred to the web version of

this article.)

S. Banerjee, I. Bose / Journal of Theoretical Biology 273 (2011) 188–196 193

are described by

dðJZÞ

dt¼

a1

1þm1JZ�

JA:JZ

JAþK2þ JA:JZ�d1JZþxðtÞ ð19Þ

dðJAÞ

dt¼

a2

1þm2JZ�d2JA ð20Þ

The noise term xðtÞ approximates the actual fluctuations whichoccur in the system. We assume the noise to be of the Ornstein–Uhlenbeck (OU) type (colored noise) (van Kampen, 1992;Chandrasekhar, 1943) as recent studies show that OU noise playsa significant role in gene expression dynamics (Suel et al., 2006;Rosenfeld et al., 2005). The noise variable has zero mean and an

exponentially decaying correlation in time, i.e.,

/xðtÞS¼ 0 ð21Þ

/xðtÞxðtuÞS¼Dlexpð�ljt�tujÞ ð22Þ

where D is the noise strength and l the inverse of the correlationtime t. Since the mean of the noise term /xðtÞS¼ 0, the variance attime t is /x2

ðtÞS¼D. The correlation time t is a measure of theaverage lifetime of a fluctuation. We use the numerical simulationalgorithm developed for the OU process (Fox et al., 1998; Gillespie,1996) to obtain solutions of the stochastic differential equations(19) and (20). The major steps of the algorithm are as follows. Let

Fig. 8. Transient pulse of JA versus time in the reduced model defined by Eqs. (15)

and (16). JA and time are dimensionless in the reduced model.

Fig. 9. Variation of JA (black line) and JZ (red line) versus time in dimensionless

units. The inset of the figure shows the mean values of JA and JZ over time with

average taken over ten individual time traces. (For interpretation of the references to

color in this figure legend, the reader is referred to the web version of this article.)

S. Banerjee, I. Bose / Journal of Theoretical Biology 273 (2011) 188–196194

x¼ JA and y¼ JZ. Knowing the values of x and y at time t, the values attime tþDt (Dt is small) are

xðtþDtÞ ¼ xðtÞþpDt ð23Þ

yðtþDtÞ ¼ yðtÞþqDt ð24Þ

where

p¼ f ðx,yÞþxðtÞ ð25Þ

q¼ gðx,yÞ ð26Þ

The functions f(x,y) and g(x,y) are, from Eqs. (19) and (20),

f ðx,yÞ ¼a1

1þm1y�

x:y

xþK2þx:y�d1y ð27Þ

gðx,yÞ ¼a2

1þm2y�x ð28Þ

The noise variable xðtþDtÞ is

xðtþDtÞ ¼ xðtÞmxþsxn1 ð29Þ

with

mx ¼ e�lDt ð30Þ

s2x ¼Dlð1�m2Þ ð31Þ

and n1 is a normal random variable with mean zero and variance 1.Through iterative computation, using Eqs. (23)–(31), the values of x

and y at successive time steps Dt can be determined. Fig. 9 showsthe time traces of x(JA) and y(JZ) for the parameter values the sameas in the cases of Figs. 7 and 8. The initial values of x and y are zero,1=l¼ t¼ 0:5 and D¼0.0625. The incremental time stepDt¼ 0:001.Fig. 9 shows the anticorrelation between JA and JZ values, the JZ

value dips when JA rises up to the peak pulse value. Fluctuations ofsufficient strength in JZ levels can give rise to transient JA pulses.

We further probe the stochastic dynamics of the full model withthe help of computer simulation based on the Gillespie (1977)algorithm. The reactions are 15 in number and are given by

Mþ JZ-M�JZ ð32Þ

M�JZ-Mþ JZ ð33Þ

JAþ IL-JAIL ð34Þ

JAIL-JAþ IL ð35Þ

JZþ JAIL-JA�JAIL ð36Þ

JZ�JAIL-JZþ JAIL ð37Þ

JZ�JAIL-JAIL ð38Þ

G1�-JZ ð39Þ

G2�-JA ð40Þ

JZ-f ð41Þ

JA-f ð42Þ

G1þM-G1� ð43Þ

G1�-G1þM ð44Þ

G2þM-G2� ð45Þ

G2�-G2þM ð46Þ

The different symbols are explained in Section 2. f denotes theproteins degradation product. G1 (G2) and G1n (G2n) are theinactive and active states respectively of the gene synthesizingJAZ proteins (JA). The active state of the gene is attained on thebinding of the transcription factor MYC2 to the regulatory region ofthe gene. For simplicity, only one JA biosynthesis gene is assumed.The stochastic rate constants, associated with the reactions, areC(i), i¼1,y,15 in appropriate units. Fig. 10 shows the variation ofthe number of JA (black curve) and JAZ (red curve) molecules as afunction of time. The values of the stochastic rate constants areC(1)¼0.2, C(2)¼0.001, C(3)¼0.1, C(4)¼0.1, C(5)¼0.1, C(6)¼1.0,C(7)¼1.0, C(8)¼12.0, C(9)¼10.0, C(10)¼0.1, C(11)¼0.4,C(12)¼1.0, C(13)¼1.0, C(14)¼1.0 and C(15)¼1.0. The stochastictime evolution of JA shows a transient pulse of large amplitudefollowed by a series of small amplitude transient pulses. As in thecase of the reduced model, the JA and JZ values are anticorrelated. InFigs. 9 and 10, a prominent JA pulse is accompanied by ‘‘secondarypulses’’ of smaller amplitude which are in reality fluctuationsaround the stable steady state representing the low JA level. Thetransient JA pulse is generated only when the JA signaling pathwayis activated. In the absence of an activating signal, the TF MYC2 iskept inactive by the JAZ repressor so that the JA biosynthesis genes

Fig. 10. Number of JA (black curve) and JZ (red curve) molecules versus time in

hours. The values of the stochastic rate constants are mentioned in the text. (For

interpretation of the references to color in this figure legend, the reader is referred to

the web version of this article.)

S. Banerjee, I. Bose / Journal of Theoretical Biology 273 (2011) 188–196 195

are no longer expressed. In a stochastic description, the low JA levelexhibits fluctuations around an average value. A reduction influctuation amplitudes may occur due to additional features notincluded in the model of the signaling pathway considered by us.There is some evidence that the MYC2 autorepresses its ownsynthesis (Farmer, 2007) a possible consequence of which is toreduce fluctuations in MYC2 levels which in turns leads to lessnoisy JA levels.

4. Discussion

The JA signaling pathway is activated in defending plantsagainst attacks by pathogens and wounding by animals. Recentexperiments (Farmer et al., 2003; Chini et al., 2007; Thines et al.,2007) provide important new knowledge on the molecular con-nectivity of the pathway. The observations suggest a model inwhich two feedback loops, one positive and the other negative,control the dynamics of JA response in the form of a transient pulse.In this paper, we propose a minimal mathematical model whichcaptures the essential features of the JA signaling pathway. To ourknowledge, no other mathematical model of the dynamics of thecoupled feedback loops in the JA signaling pathway has beenproposed so far. The model demonstrates the formation of thetransient JA pulse for different parameter values. The pulse is anoutcome of two opposing influences mediated via the positive andnegative feedback loops. We compute the dependence of the pulseamplitude, duration and peak time on the key parameters of thetheoretical model. The full mathematical model, described in termsof Eqs. (4)–(8), can be reduced to a two-variable model (Eqs. (15)and (16)) for which the physical steady state solution is obtainedanalytically. There is only one such solution, corresponding to lowJA levels, which is stable in an extended region of the parameterspace. A similar conclusion holds true for the full dynamical model.We explore the dynamics of both the full and reduced models usingdeterministic as well as stochastic formalisms. The TF MYC2activates the expression of both the JA biosynthesis and JAZ genes.In normal circumstances, the expression activity is low as JAZproteins repress MYC2 by forming bound complexes with the TFproteins. For the JA pulse to be initiated, free MYC2 is required.Once the JA synthesis is initiated, the positive feedback loop rampsup further JA production. The negative feedback loop acts against

the enhancement so that the JA level attains a steady lower valueafter a certain time interval. JA-mediated destruction of JAZrepressors frees MYC2 which in turn activates JAZ gene expression.The production of JAZ proteins completes the negative feedbackloop through the repression of MYC2 activity (Fig. 1). In thedeterministic formalism, the transient JA pulse is obtained whena certain amount of free MYC2 is available at time t¼0. Theavailability of free MYC2 is JA-dependent. We do not model theexplicit steps of JA biosynthesis and the synthesis of MYC2. Westudy the dynamics of the JA signaling pathway with free MYC2 asinput. The central role of MYC2 in the model is dictated by availableexperimental evidence but there could be other TFs playing similarroles (see Pieterse et al., 2009, Fig. 5(b)). The prominent role ofMYC2 could be tested experimentally by adding free MYC2 to plantleaves and checking whether transient JA pulses are obtained evenin the absence of wounding. In the stochastic case, fluctuations inthe numbers of JA and JAZ proteins can activate pulse formation.This is clearly understood in the framework of the reduced model.As Fig. 9 distinctly shows, the fluctuations in JA and JAZ numbersare anticorrelated. A temporary dip in the JAZ levels results in theincreased availability of free MYC2 and the subsequent generationof a JA pulse. Single cell experiments on a variety of organisms haveestablished the stochastic nature of gene expression (Kaern et al.,2005; Raj and van Oudenaarden, 2008) and it is now commonknowledge that protein fluctuations have a considerable effect oncellular processes.

The operating principle behind transient gene expressionactivity in the JA signaling pathway is based on a combination ofpositive and negative feedback loops and the availability of freeMYC2 to transactivate the expression of the JA biosynthesis and JAZgenes. Due to the transient nature of the JA pulse, individual cellsattain a state in which elevated levels of JA induce the expression oftarget genes involved in defense response and after a time intervalreturn to the quiescent state. In B. subtilis, a core module consistingof a fast positive and a slow negative feedback loop can explaintransient gene expression activity resulting in entry into thecompetent state and subsequent exit from it (Suel et al., 2006).The system exhibits excitable dynamics with three fixed points:one stable at low ComK concentration (the vegetative state), oneunstable saddle node at intermediate ComK level and an unstablespiral node at high ComK concentration (the competence state)(Suel et al., 2006). Transient activation to the competence stateoccurs if the ComK level crosses the intermediate fixed point due tofluctuations originating in stochastic gene expression. In contrast,the dynamics of JA pulse formation involve only one fixed point. Asimilar situation prevails in the case of the HIV regulatory circuit. Aminimal model of the circuit dynamics predicts a single fixedpoint which is stable provided a specific condition is satisfied(Weinberger, 2007). In this case, the long-lived pulse of Tat proteinsresults from transient positive feedback amplification and deacti-vation via stronger back reactions like deacetylation of the Tatprotein and unbinding of the regulatory molecules. The backreactions opposing the action of the positive feedback loop havean effective influence similar to that of a negative feedback loop.There is now experimental validation of the proposal that noise ingene expression determines the cell fate (competence versus non-competence) in B. subtilis (Maamar et al., 2007). In the Tat circuitalso, stochastic molecular fluctuations intrinsic to gene expressioncan activate transient Tat pulses (Weinberger, 2007). In the case ofthe JA signaling pathway, noise aids in the activation of the JA pulsethough the off-state (low JA level) is found to be quite stable tofluctuations (Figs. 9 and 10). The negative feedback enhances theoff-state stability in this case.

An interesting aspect of JA-response not addressed in thepresent study relates to memory formation of past woundingevents or attacks priming plants for a more effective JA-mediated

S. Banerjee, I. Bose / Journal of Theoretical Biology 273 (2011) 188–196196

defense response to subsequent attacks (Galis et al., 2009; Storket al., 2009). Evidence of memory formation could lie in themodification of (1) the peak time of JA accumulation, (2) amplitudeof JA pulse or (3) baseline of accumulated JA due to discrete JAbursts elicited by repeated attacks (Galis et al., 2009). The negativefeedback loop b in the JA-signaling network constitutes a well-known motif in gene regulatory networks. A negative feedback loopby itself may give rise to oscillatory dynamics or alternativelypulsed protein production (Novak and Tyson, 2008; Batchelor et al.,2009). The p53–MDM2 loop provides a well-known example ofnegative feedback (Novak and Tyson, 2008). The loop is function-ally active in response to DNA damage. Single cell experimentsshow that on DNA damage by g�irradiation a series of undampedp53 pulses of fixed amplitude and duration is generated. Theamount of g�irradiation determines the number of pulses but hasno effect on the amplitude and duration of the pulses. Thefunctional response of the MYC2-JAZ negative feedback loop inthe JA signaling pathway to wounding signals of varying magnitudeis worth exploring at the single cell level to gain insight on keydynamical features. It is generally difficult to link circuit dynamicswith circuit structure. The identification of feedback loops in the JAsignaling pathway allows for the construction of a minimalmathematical model which correctly reproduces the formationof JA pulses on activation. The deterministic and stochastic originsof pulse formation are obtained through computational analysis.The dependence of key pulse parameters on the kinetic rateconstants is demonstrated quantitatively. Further experimentsare needed to validate the theoretical predictions as well as toelucidate the operating principle of the JA signaling pathway at amicroscopic level. Comparative studies of operating principlescontrolling transient gene expression activity in diverse organismsare expected to provide insight on the common mechanismsorganisms living in a complex environment adopt for respondingto stress signals.

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