Emergent memory in cell signaling: Persistent adaptive dynamics in cascades can arise from the diversity of relaxation time-scales
EEmergent memory in cell signaling: Persistent adaptive dynamics in cascades canarise from the diversity of relaxation time-scales
Tanmay Mitra , , Shakti N. Menon and Sitabhra Sinha , The Institute of Mathematical Sciences, CIT Campus, Taramani, Chennai 600113, India. Homi Bhabha National Institute, Anushaktinagar, Mumbai 400 094, India. (Dated: January 15, 2018)The mitogen-activated protein kinase (MAPK) signaling cascade, an evolutionarily conserved mo-tif present in all eukaryotic cells, is involved in coordinating critical cell-fate decisions, regulatingprotein synthesis, and mediating learning and memory. While the steady-state behavior of thepathway stimulated by a time-invariant signal is relatively well-understood, we show using a com-putational model that it exhibits a rich repertoire of transient adaptive responses to changes instimuli. When the signal is switched on, the response is characterized by long-lived modulations infrequency as well as amplitude. On withdrawing the stimulus, the activity decays over timescalesmuch longer than that of phosphorylation-dephosphorylation processes, exhibiting reverberationscharacterized by repeated spiking in the activated MAPK concentration. The long-term persis-tence of such post-stimulus activity suggests that the cascade retains memory of the signal for asignificant duration following its removal, even in the absence of any explicit feedback or cross-talkwith other pathways. We find that the molecular mechanism underlying this behavior is relatedto the existence of distinct relaxation rates for the different cascade components. This results inthe imbalance of fluxes between different layers of the cascade, with the repeated reuse of activatedkinases as enzymes when they are released from sequestration in complexes leading to one or morespike events following the removal of the stimulus. The persistent adaptive response reported here,indicative of a cellular “short-term” memory, suggests that this ubiquitous signaling pathway playsan even more central role in information processing by eukaryotic cells.
PACS numbers: 87.16.Xa,87.17.Aa,87.18.Vf
I. INTRODUCTION
Intra-cellular signaling networks are paradigmatic ex-amples of complex adaptive systems that exhibit a richrepertoire of responses to stimuli [1]. Such networks me-diate the response of a cell to a wide variety of extra-and intra-cellular signals primarily through a sequenceof enzyme-substrate biochemical reactions [2, 3]. Whilethe complexity of the entire signaling system is daunt-ing [4], it is possible to gain an insight into how it func-tions by focusing on a key set of frequently occurringmotifs. These often take the form of linear signaling cas-cades, referred to as pathways. One of the best knownof these pathways is the mitogen-activated protein ki-nase (MAPK) cascade that is present in all eukaryoticcells [5, 6]. It is involved in regulating a range of vital cel-lular functions, including proliferation and apoptosis [6],stress response [7] and gene expression [8]. This signal-ing module comprises a sequential arrangement of threeprotein kinases, viz., MAPK, MAPK kinase (MAP2K)and MAPK kinase kinase (MAP3K). Modular function isinitiated when extracellular signals stimulate membrane-bound receptors upstream of the cascade, with the in-formation being relayed to MAP3K by a series of inter-mediaries. Activated kinases in each layer of the modulefunction as enzymes for phosphorylating (and thereby ac-tivating) the kinase in the level immediately downstream,with the subsequent deactivation being mediated by cor-responding dephosphorylating enzymes known as phos-phatases (P’ase). The terminal kinase in this cascade, i.e., MAPK, transmits the signal further downstream byphosphorylating various proteins including transcriptionregulators [9]. Extensive investigations into the steady-state behavior of the cascade have contributed towards anin-depth understanding of several emergent features in-cluding ultrasensitivity [10], and oscillations [11, 12] thatarise through retrograde propagation of activity [13, 14].One of the striking features of the cascade is the occur-rence of bistability which allows the system to switch be-tween two possible states corresponding to low and highactivity [12, 15–18]. This provides a post-transcriptionalmechanism for obtaining a sustained response from tran-sient signals, i.e., cellular memory [19, 20].Memory can be understood as long-term alterationsin the state of a system in response to environmen-tal changes, which allow the system to retain informa-tion about transient signals long after being exposed tothem [19]. This can arise in the cell through mech-anisms such as auto-regulatory transcriptional positivefeedback [21] and nucleosomal modifications [22]. Inthe context of cell-fate determination, it has been shownthat an irreversible biochemical response can be gener-ated from a short-lived stimulus through feedback-basedbistability [20]. This corresponds to a permanent alter-ation of the state of the system, thereby actively main-taining ‘memory’ of the signal. As bistability has alsobeen observed to arise through multi-site phosphoryla-tion in signaling modules, protein phosphorylation hasbeen suggested as a plausible post-transcriptional mech-anism for cellular memory [19, 23, 24]. In particular,there have been extensive investigations of the MAPK a r X i v : . [ q - b i o . S C ] J a n cascade as it integrates a large range of signals receivedby the cell in order to control numerous cellular deci-sions [25–30]. While these investigations have consideredthe steady state behavior of the system, one may also ob-serve transitory modulations in the response of the cas-cade in a changing environment. The latter could encodeinformation about prior stimuli to which the system wasexposed, and can be a potential mechanism for impartinga form of “short-term” memory to the signaling cascade.In this paper we show that a linear MAPK cascadecan indeed exhibit short-term memory through transientmodulations in its response to an environmental change.Crucially, this can arise even in the absence of explicitfeedback between different layers or cross-talk with otherpathways. These modulations can persist long after theinitial trigger, lasting for durations that are several or-ders of magnitude longer than the time-scales associatedwith phosphorylation-dephosphorylation processes. Wedemonstrate that this occurs both when a signal beginsactivating the MAPK cascade, as well as when it is with-drawn. On application of the stimulus, the module ex-hibits long-lived frequency and amplitude modulations inthe activation profile of the constituent kinases. Follow-ing the withdrawal of stimulus, activity in the cascade de-cays over an extremely long time-scale, during which re-verberatory dynamics, characterized by large-amplitudespiking in MAP Kinase activity, can be observed. Weexplain the emergence of such long-lived memory of thewithdrawn stimulus in terms of the imbalance of fluxesbetween different layers of the cascade, which results fromthe diversity of relaxation time-scales of the cascade com-ponents, and the reuse of activated kinases as enzymeswhen they are released from sequestration. This phe-nomenon is seen to be robust with respect to variationsin the molecular concentrations of the constituent kinasesand phosphatases. Our results reveal that a biochemicalsignaling module as simple as the MAPK cascade is ca-pable of exhibiting short-term memory that is manifestedas persistent modulations in the adaptive response of thesystem to changes in stimuli. II. METHODS
The dynamics of the three layer MAPK signalingcascade has been simulated using the Huang-Ferrellmodel [10]. Each of the constituent kinase andphosphatase-mediated enzyme-substrate reactions com-prise (i) a reversible step corresponding to the forma-tion of the enzyme-substrate complex and (ii) an ir-reversible product formation step corresponding to theactivation/deactivation of a kinase, as described in theSupplementary Information. The time-evolution of themolecular concentrations of the different components ofthe cascade are modeled using a set of coupled ordinarydifferential equations (see Supplementary Information)that are integrated using the stiff solver ode15s imple-mented in
MATLAB Release 2010b . Note that the quasi- steady-state hypothesis has not been invoked [31]. To en-sure that initially all kinases are non-phosphorylated weprepare the initial resting state of the system by simulat-ing it for a long duration ( ∼ mins) in the absence ofany signal. Subsequently MAP3K is exposed to a stimu-lus of amplitude S and duration 5000 minutes. Followingthe removal of the stimulus, we continue to simulate thesystem until it returns to the resting state or the simula-tion duration exceeds 10 minutes.We have analyzed the long-lived reverberatory activityof the cascade after the removal of the stimulus by usingthe following measures:(i) The primary recovery time ( τ PR ). Following the ac-tivation of the cascade by introducing a stimulus, themaximum concentration R max of MAPK ∗∗ is recorded(Note that ∗∗ represents a doubly phosphorylated kinasewhile ∗ indicates that it is singly phosphorylated). Onremoving the stimulus, MAPK activity starts to decay.The time taken for MAPK ∗∗ to monotonically decreaseto half of R max is defined as the primary recovery time( τ PR ).(ii) Number of spikes during relaxation ( N r ). Followingprimary recovery, MAPK activity may exhibit a seriesof spikes, which are defined to be occurring wheneverMAPK ∗∗ concentration exceeds 70% of R max . The num-ber of such spikes that are observed before the cascadereaches its resting state is designated as N r .(iii) The total duration of reverberatory activity ( τ r ).When spiking is observed in MAPK activity followingthe removal of the applied stimulus, the reverberatoryactivity duration is defined as the interval between thetermination of primary recovery and the final spike event,i.e., τ r = t final − τ P R . The time of the i th spike t i is de-fined as the instant when MAPK activity reaches maxi-mum during that particular event. For τ P R > minutes.(iv) The total memory time ( τ m ). The total duration ofmemory activity following removal of the applied stimu-lus is defined as the sum of the primary recovery timeand the total duration of reverberatory activity, i.e., τ m = τ P R + τ r . Note that when the steady-state behaviorof the cascade in presence of the signal is oscillatory, onwithdrawing the signal the activity may decay extremelyrapidly resulting in τ m ≈ Relaxation time ( τ x ). For the situations where thesteady state corresponds to a fixed-point attractor wedefine a relaxation time τ x for each constituent of thecascade. This is the time required by its concentrationto evolve to the half-way point between the resting stateand steady state values. III. RESULTS
Emergence of persistent modulations in kinase activ-ity.
For the results reported in this paper we consider
FIG. 1: Adaptive response of MAPK cascade to a chang-ing stimulus. (a) Schematic representation of a linear MAPKcascade comprising three layers. Signaling is initiated by astimulus S activating MAPK kinase kinase (MAP3K). Activa-tion/deactivation of kinases is achieved by adding/removingphosphate groups, which is referred to as phosphoryla-tion/dephosphorylation respectively. The activated MAP3Kregulates the phosphorylation of MAPK kinase (MAP2K).Doubly phosphorylated MAP2K, in its turn, controls the ac-tivation of MAPK. The response of the cascade to the signal ismeasured in terms of MAPK activity, viz., the concentrationof doubly phosphorylated MAPK. Deactivation of a phospho-rylated kinase is regulated by the corresponding phosphatase(indicated by P’ase) in the corresponding layer of the cascade.The numbers 1 − S . Withdrawing the stimulus can result in persistent large-amplitude spiking in the response of MAPK, suggestive ofa form of “short-term” memory. The maximum response ofMAPK to the stimulus is denoted by R max . The primary re-covery time ( τ PR ) is characterized as the duration followingwithdrawal of stimulus after which MAPK activity decreasesto its half-maximum value ( R max /
2) for the first time. Theduration over which reverberatory dynamics occurs is indi-cated by τ r , while the total duration for which memory of thewithdrawn stimulus persists is τ m = τ PR + τ r . the Huang-Ferrell model of the MAPK signaling cas-cade [10], schematically shown in Fig. 1 (a). Typically,investigations into the dynamics of this model focus onthe steady-state response to sustained stimulation. Incontrast, here we report on the transient activity of thesystem following a change in the stimulus. Specifically,we describe the response immediately following the in-troduction of a signal of amplitude S and that followingits removal. Our results reveal that such transients canbe unexpectedly long-lived, lasting for durations that aremuch longer compared to the time-scales associated withthe phosphorylation and dephosphorylation processes inthe cascade (Fig. 1, b).We first report the behavior of a cascade that is ini-tially in the resting state (characterized by the absenceof any phosphorylated components) when it is exposedto a signal. The transient activity that immediately fol-lows exhibits several non-trivial features such as regularspiking in the activity of MAP2K and MAPK dependingon the total concentrations of the kinases (Fig. 2, b-e)and the signal strength. For a fixed initial state and sig-nal strength, the spikes can further show modulation intheir frequency (Fig. 2, c-e) as well as amplitude (Fig. 2,b and d). In certain cases, both types of modulationcan be observed (Fig. 2, d). In the representative timeseries of MAPK activity shown in Fig. 2(a-e), the sys-tem dynamics eventually converges to a stable fixed point(Fig. 2, a-d) or a stable limit cycle (Fig. 2, e), with theattractors being independent of initial conditions (corre-sponding phase space projections are shown in Fig. 2, f-i).Note that when phosphorylated components are initiallypresent, the system reaches the asymptotic state faster.When the signal is withdrawn, the signaling cascadecan respond with large-amplitude spiking behavior in theMAPK activity before eventually relaxing to the restingstate (Fig. 3). These phenomena are seen for a range ofstimuli strengths and are indicative of a form of memorythat can be achieved without explicit feedback or inter-pathway crosstalk. An essential condition for observingthe reverberatory activity is that prior to withdrawingthe applied stimulus, the system state has been drivenabove the low-amplitude response regime. The complexmodulations seen in these figures may arise as a resultof coexisting attractors. For example, in Fig. 2 (d) thesystem state spends considerable time in the basin ofattraction of a limit cycle before approaching a stablefixed point (see Supplementary Information for details). Processes underlying long-lived memory and reverber-atory dynamics.
When the stimulus is withdrawn fromthe MAPK cascade, the decline in MAP Kinase activitycomes about through MAPK ∗∗ binding to MAPK P’asewhich dephosphorylates it, resulting in an increased con-centration of MAPK ∗ [Step 1, Figs. 1(a) and 4(a)]. Inturn, the phosphatase binds to MAPK ∗ thereby deacti-vating it to MAPK which results in an extremely rapiddecline in the concentration of MAPK ∗ (Step 2). Concur-rently, the deactivation of MAP2K ∗∗ is delayed, as mostof it is bound in the complex MAP2K ∗∗ .MAPK that has FIG. 2: Transient activity in MAPK cascade immediatelyfollowing the application of a stimulus having amplitude S = 1 . × − µM at t = 0. (a-e) Characteristic time se-ries for the normalized concentration of doubly phosphory-lated MAPK ( n K ∗∗ ) shown for different total concentrationsof kinases. The concentration of active MAPK is insignificantprior to the time periods shown in panels (a-e). (f-j) Trajecto-ries representing the evolution of the systems in panels (a-e)in the projection of the phase-space on the planes comprisingnormalized concentrations of active MAP3K ( n K ∗ ), singlyphosphorylated MAP2K ( n K ∗ ) and active MAPK ( n K ∗∗ ).The concentrations have been normalized by the total concen-tration of MAP3K ([3 K ] tot ), MAP2K ([2 K ] tot ) and MAPK([ K ] tot ), respectively. The light blue and dark blue markers ineach of the panels (f-j) demarcate the portion of the trajecto-ries that correspond to the time series shown in panels (a-e).The steady state of the system is represented by a red markerin panels (f-i). In panels (e) and (j), the system convergesto a stable limit cycle. For details of parameter values forthe systems shown in each of the panels see SupplementaryInformation. a long time-scale of disassociation. To proceed furtherwe can analyze the constituent processes in terms of thenormalized chemical flux N Flux of a molecular species,i.e., its rate of growth expressed relative to the maxi-mum rate of growth of MAPK ∗∗ . We observe that thesuppression of MAP2K ∗∗ deactivation mentioned aboveresults in its normalized chemical flux exceeding that ofMAPK [Fig. 4(b)]. Thus, there is a net growth in activ-ity in the MAP Kinase layer as whenever MAP2K ∗∗ isreleased from the complex, it is available to phosphory- FIG. 3: Transient activity in MAPK cascade immediately fol-lowing the withdrawal (at t = 0) of an applied stimulus hav-ing amplitude S = 1 . × − µM . (a-e) Characteristic timeseries for the normalized concentration of doubly phosphory-lated MAPK ( n K ∗∗ ) shown for different total concentrationsof kinases. (f-j) Trajectories representing the evolution of thesystems in panels (a-e) in the projection of the phase-spaceon the planes comprising normalized concentrations of activeMAP3K ( n K ∗ ), singly phosphorylated MAPK ( n K ∗ ) andactive MAPK ( n K ∗∗ ). The concentrations have been normal-ized by the total concentration of MAP3K ([3 K ] tot ), MAP2K([2 K ] tot ) and MAPK ([ K ] tot ), respectively. The steady stateof the system prior to the withdrawal of the stimulus is rep-resented by a red marker (panels f-i). The system in panels(e) and (j) is seen to relax from a state characterized by sta-ble limit cycle oscillations (represented by the blue marker).In each trajectory shown in (f-j) the grey marker denotes thestate of the system corresponding to the final time point inpanels (a-e). The concentration of active MAPK is close to itsresting state value following the time period shown in (a-e).The parameter values for each panel are same as those for thecorresponding panels in Fig. 2. late MAPK which results in an increase in the concentra-tion of MAPK ∗ (Step 3). The resulting rise in MAPK ∗ manifests as a spike in its concentration [Fig. 4(a)], andit subsequently gets phosphorylated again to increaseMAPK ∗∗ concentration even in the absence of any stim-ulation (Step 4). When the net difference between thenormalized flux of MAP2K ∗∗ and MAPK reaches a max-imum, the normalized chemical flux of MAPK ∗∗ attainsits highest value and consequently peak activity of MAP FIG. 4: Processes underlying emergent memory and reverber-atory dynamics in the MAPK cascade. (a) A characteristictime-series for the normalized concentrations of singly anddoubly phosphorylated MAPK ( n K ∗ and n K ∗∗ , respectively)following the removal of an applied stimulus of amplitude S = 2 . × − µM at t = 0. The numbers (1 −
4) representthe sequence of events that lead to the emergence of the post-stimulus large-amplitude spiking activity shown schematicallyin Fig. 1 (b). (b) Normalized chemical flux N Flux of MAPKand MAP2K ∗∗ shown for the segment of the time-series wherethe spiking behavior in n K ∗∗ is observed following the with-drawal of the stimulus to MAP3K [demarcated by brokenvertical lines in (a)]. (c) Normalized chemical flux N Flux ofMAPK ∗∗ shown along with the difference between the nor-malized fluxes of MAP2K ∗∗ and MAPK for the duration in-dicated by broken vertical lines in (b) corresponding to thepeak in the spiking activity of MAPK ∗∗ . For both panels (b)and (c), normalization of flux is with respect to the maximumof the flux for MAPK ∗∗ . (d) Characteristic time-series for thereverberatory activity of MAPK following the withdrawal ofa stimulus of amplitude S = 1 . × − µM at t = 0, showingthe normalized concentration of MAPK ∗∗ ( n K ∗∗ ) along withthat of the protein complex MAP2K ∗∗ .MAPK ( n K ∗∗ .K =[MAP2K ∗∗ .MAPK]/[2 K ] tot ). The reference line shows thatthe peak normalized concentration of the protein complexeventually decreases over time. For details of parameter val-ues for (a-c) see Supplementary Information. The parametervalues for panel (d) are same as those in Fig. 3(d). The steadystate of the system prior to the withdrawal of the stimulus isrepresented by a red marker [panels (a) and (d)] while the greymarker in (d) corresponds the final time point in Fig. 3(d). Kinase is observed [Fig. 4(c)]. Thus, steps 1-4 representone complete cycle of MAP Kinase reverberatory activ-ity characterized by an initial decline and a subsequentrise in MAPK ∗∗ concentration. These steps are subse-quently repeated a number of times resulting in a series ofspikes in MAPK activity [Fig. 4(d)]. The abrupt natureof the rise and fall of MAP Kinase activity that mani-fests as spikes is a consequence of the bistable nature ofthe dynamics in the MAPK layer of the cascade [15, 17].We note that similar spiking behavior is also observed in FIG. 5: Components of the MAPK cascade exhibit relaxationbehavior occurring over a broad range of time-scales. Decayof activity is shown after withdrawing an applied stimulus ofamplitude S = 1 . × − µM . (a) The relaxation times τ x of the different molecular species (non- , singly- and doublyphosphorylated kinase proteins) in each of the layers of thecascade vary with the total concentration of MAP2K. Thenature of this dependence is distinct for lower and higher val-ues of [2 K ] tot , which is most prominently observed in thelower layers of the cascade. (b) The occurrence of distinctregimes in the relaxation behavior of MAPK ∗∗ for different[2 K ] tot is related to the corresponding increase in the steadystate value attained by MAPK ∗∗ concentration under sus-tained stimulation of the cascade. At a specific value of thesteady-state normalized MAPK activity n K ∗∗ , we observe acrossover from the regime characterized by slowly increasing τ x seen at lower total concentrations of MAP2K to a regimewhere τ x increases relatively rapidly for higher [2 K ] tot . (c)The crossover behavior is also observed in the dependence ofthe closely related measure τ PR , the primary recovery time(see Methods), on [2 K ] tot . The difference between the tworegimes become more prominent upon increasing the totalconcentration of MAP3K ([3 K ] tot ). For both panels (a) and(b) [ K ] tot = 0 . µM and [3 K ] tot = 2 . nM , while for panel (c),[ K ] tot = 0 . µM . For details of all other parameter values seeSupplementary Information. the activity of MAP2K, with the phase of the MAP2K ∗∗ spikes shifted slightly forward with respect to the cor-responding ones in MAPK ∗∗ , which suggests that theyresult from retrograde propagation of activity from theMAPK to the MAP2K layer [14]. On the other hand,MAP3K shows a monotonic decline in its activity follow-ing the removal of the stimulus.In order to characterize in detail the memory of prioractivity retained by the cascade which is manifested aslong-lived transient reverberations following the with-drawal of stimulus, we use the following measures (seeMethods): (i) the primary recovery time ( τ PR ), (ii) thenumber of spikes ( N r ) that occur during the relaxation FIG. 6: Dependence of reverberatory activity on the total ki-nase concentrations, viz., [ K ] tot , [2 K ] tot and [3 K ] tot . (a-b)The number of spikes N r , (c-d) the total memory time τ m (in minutes) and (e-f) isosurfaces for N r observed on with-drawing an applied stimulus of amplitude S [= 0 . × − µM for (a,c,e) and 1 . × − µM for (b,d,f)] are shown as func-tions of total concentrations of the three kinases. (g) Theprimary recovery time τ PR (stars) and the total duration ofreverberatory activity τ r (filled circles) are shown for differ-ent values of N r (indicated by the color bar). While τ PR increases monotonically with increasing total MAPK concen-tration, τ r shows a more complex dependence ([2 K ] tot = 3 µM and [3 K ] tot = 4 nM ). (h) The dependence of τ r on [ K ] tot fordifferent values of N r has a similar nature for different choicesof [3 K ] tot (indicated by the color bar, [2 K ] tot = 3 µM ). Notethat for panel (h), we consider only situations where the sys-tem attains a steady state on maintaining stimulation. Fordetails of all other parameter values see Supplementary Infor-mation. process, (iii) the temporal intervals between successivespikes ( t i − t i − , where t i is the time of occurrence of the i th spike event) and (iv) the total duration of reverber-atory activity ( τ r ) following primary recovery. The total memory time ( τ m ) is the sum of τ PR and τ r as indicatedin Fig. 1 (b). In the following we use these measuresto present a detailed characterization of the behavior ofthe cascade components over a range of parameter values(Figs. 5-7). MAP Kinase cascade components have different recov-ery timescales.
As mentioned earlier, the emergence oflong-lived reverberatory activity of MAPK following thewithdrawal of an applied stimulus can be linked to theflux imbalance of different cascade components whichsuggests significant differences in their rates of relaxation.As shown in Fig. 5 (a), this is indeed the case, even forparameter regimes where no spiking activity of MAPKis observed (i.e., N r = 0). As can be seen, the natureof increase of the relaxation time with increasing totalconcentrations of kinase protein MAP2K is distinct forthe different molecular species and also depends on thestate of their phosphorylation. In the lower layers of thecascade, we also find a crossover between two regimesseen at lower and higher values of [2 K ] tot respectively.These regimes are characterized by relatively slow andrapid increases (respectively) in the recovery times withincreasing [2 K ] tot , and appear to be related to the steady-state value attained by MAPK activity upon sustainedstimulation of the cascade for the corresponding valueof [2 K ] tot [Fig. 5 (b)]. The crossover between the tworegimes is seen to occur for a value of [2 K ] tot for which ∼
17% of MAPK is activated for the parameter valuesused in Fig. 5 (b).The distinct regimes are also observed in the de-pendence of the primary recovery time τ PR on [2 K ] tot [Fig. 5(c)]. As can be observed, the difference betweenthe regimes becomes more pronounced with an increasein the total concentration of MAP3K. An importantpoint to note is that for lower values of [2 K ] tot , the re-covery time decreases with increasing [3 K ] tot while thereverse trend is seen for higher values of [2 K ] tot . Wehave verified that increasing the stimulus amplitude S while keeping the total MAP3K concentration fixed hasa similar effect on the relaxation behavior of activatedMAPK (see Supplementary Information). As increas-ing total concentration of MAP2K results in increasedsteady-state activity of MAPK, we conclude that, in gen-eral, higher activity states of MAPK are associated withincreasing relaxation time when either the signal or thesubstrate (MAP3K) is increased. Conversely, for statescharacterized by much lower MAPK activity, larger val-ues of S or [3 K ] tot results in reduced relaxation periods. Dependence of reverberatory activity on total kinaseconcentrations.
Diverse cellular environments are char-acterized by different total concentrations of the vari-ous molecular components of the MAPK cascade. Thus,in order to determine the robustness of spiking and re-verberatory activity following the removal of an appliedstimulus, it is important to see how they are affected byvarying total kinase concentrations. Such a study willalso indicate the ease with which these phenomena canbe experimentally observed. Fig. 6 shows the variation ofdifferent measures of reverberatory activity on the totalconcentrations of MAPK, MAP2K and MAP3K. Whilethere is a complex dependence on these parameters forthe exact number of spikes N r and the duration of thetotal memory time τ m , the phenomenon of reverberatoryactivity following withdrawal of stimulation can be ob-served over a large range of the parameter space, under-lining its robustness. We also observe that on increasing[3 K ] tot , the response of N r to variation in [ K ] tot and[2 K ] tot becomes relatively homogeneous. Increasing thestimulus amplitude S [compare panels (a,c,e) with (b,d,f)of Fig. 6] does not seem to alter the qualitative natureof the variation in N r and τ m over the parameter spacein general, although we do observe that the domains cor-responding to different values of N r occupy different re-gions [Fig. 6(e and f)]. Note that for low [3 K ] tot , highvalues of N r are observed to coexist with low values of τ m [Fig. 6(a,c and b,d)]. While it may appear surprising thatthese two measures of memory are not in consonance inthis region of parameter space, it can be explained by not-ing that the stimulated system is in an oscillatory state,and following the removal of the signal these relativelyhigh-frequency oscillations cease after a short duration.Fig. 6 (g) suggests that the variation seen in τ m as afunction of the total MAPK concentration for a specific N r is mostly governed by τ r , the total duration of rever-beratory activity, with the corresponding dependence of τ P R on [ K ] tot being weak.As the total MAPK concentration is increased, we ob-serve that while the primary recovery time increases al-most linearly, the nature of the reverberatory dynamicsas reflected in τ r shows a more complex dependence on[ K ] tot [Fig. 6 (g)]. If for a given value of [ K ] tot the MAPKactivity following withdrawal of the stimulus shows N r spikes over a duration of τ r , then on increasing [ K ] tot the time-interval between the spikes increases (therebyresulting in an increase of τ r ) until a critical value beyondwhich the last of the N r spike no longer appears. Thus,at this point N r reduces by unity with a concomitantdrop in τ r . This series of events is repeated for steadilydecreasing values of N r as the total MAPK concentra-tion is increased further. Each value of N r is associatedwith a characteristic rate of increase in τ r with [ K ] tot .With a reduction in N r (as a result of increasing [ K ] tot ),this rate is found to decrease as well, which suggests asaturation of the system response. These results are ro-bust with respect to different choices of total MAP3Kconcentration as can be seen from Fig. 6(h), suggestingthat similar behavior will be seen for a range of strengthsfor the applied signal (see Supplementary Information). Dependence of reverberatory activity on total phos-phatase concentrations.
We have also investigated therole that phosphatase availability plays on the reverber-atory activity of the cascade following the withdrawalof the stimulus. As is the case for total kinase concen-trations shown in Fig. 6, we see from Fig. 7 (a-b) thatthe number of spikes N r and the duration of total mem-ory time τ m depend on the total concentrations of the FIG. 7: Dependence of reverberatory activity on the to-tal concentrations of the phosphatases MAPK P’ase ([ P K ]),MAP2K P’ase, ([ P K ]) and MAP3K P’ase ([ P K ]). (a) Thenumber of spikes N r and (b) the total memory time τ m (inminutes) observed on withdrawing an applied stimulus of am-plitude S = 0 . × − µM . Situations where the primaryrecovery time is longer than a maximum or cut-off value (seeMethods), such that the duration of the reverberatory dy-namics cannot be properly measured, are indicated by thecolor corresponding to “U”. (c) The interval between suc-cessive spikes i − i increases with time ( t i being thetime of occurrence of the i th spike). As the MAPK P’aseconcentration is increased, the durations of these intervalsare seen to increase. The total concentrations of the othertwo phosphatases are maintained at [ P K ] = 680 pM and[ P K ] = 10 pM . (d) The variation of primary recovery time τ PR (stars) and the total duration of reverberatory activity τ r (filled circles) as a function of total MAPK P’ase concen-tration are shown for different values of N r (indicated by thecolor bar). While τ PR decreases monotonically with increasing[ P K ], τ r shows a more complex dependence ([ P K ] = 200 pM and [ P K ] = 6 pM ). (e) Dependence of the total memory time τ m on total MAPK P’ase concentration ([ P K ] shown in logscale) for different total concentrations of MAP2K P’ase (val-ues indicated above each of the three panels) and MAP3KP’ase (indicated using different colors as shown in the colorbar). Note that we consider only situations where the systemattains a steady state on maintaining stimulation. For detailsof all other parameter values see Supplementary Information. phosphatases MAPK P’ase, MAP2K P’ase, and MAP3KP’ase. For larger values of the concentrations, viz., [ P K ],[ P K ] and [ P K ], respectively, the system operates in thelow-amplitude response regime. As mentioned earlier,the reverberatory MAPK dynamics during recovery fol-lowing withdrawal of the applied stimulus will not beseen in this regime. As the phosphatase concentrationsare decreased, spiking behavior of MAPK activity is ob-served with both τ m and N r attaining high values in anoptimal range. The large variation seen in τ m [Fig. 7 (b)]arises as regions in [ P K ]-[ P K ] parameter space charac-terized by the same value of N r are seen to exhibit arange of different values of τ r and τ P R [Fig. 7 (d)]. Forreverberatory activity associated with a specific N r , weobserve that the duration τ r increases with increasing to-tal MAPK P’ase concentration. This is a consequence ofthe intervals between successive spikes ( t i − t i − ) increas-ing with [ P K ] as is shown in Fig. 7 (c). Note that theresults are qualitatively similar for different amplitudesof the applied stimulus (see Supplementary Information).However, increasing [ P K ] results also in decreased timefor primary recovery τ P R [Fig. 7 (d)], which in conjunc-tion with the previously mentioned result leads to non-monotonic dependence of the total memory time τ m onphosphatase availability. While this non-monotonicity issuggested in Fig. 7 (b), it is shown clearly in Fig. 7 (e)where the central panel corresponds to situations wherespiking behavior is observed in MAPK activity. Investi-gation into the dependence of τ m on P K [Fig. 7(e)] revealsthat the range of [ P K ] over which reverberatory activity(i.e., N r (cid:54) = 0) occurs is demarcated by discontinuities inthe functional dependence of τ m on P K . For intermediate P K [Fig. 7(e), central panel] where the system attains asteady state on maintaining stimulation, the spiking ac-tivity following withdrawal of the stimulus becomes moreprominent for low total concentration of MAP3K P’ase.For higher P K [Fig. 7(e), right panel] where the sys-tem becomes oscillatory over an intermediate range of[ P K ], reverberatory activity is observed over a broaderrange of [ P K ]. While we have assumed that the samephosphatase acts on both the singly and doubly phos-phorylated forms of the kinase in a particular layer ofthe cascade (as in the canonical Huang-Ferrell model),we have explicitly verified that our results are not sensi-tively dependent on this. IV. DISCUSSION
In this paper we have shown that an isolated MAPKsignaling module can serve as a fundamental motif in theintra-cellular signaling network for imparting a form ofshort-term memory to the cell. The emergence of long-lived reverberatory activity reported here arises from thediversity of relaxation timescales for the different compo-nents of the MAP Kinase cascade, which results in fluximbalance between activation of the MAPK layer anddeactivation in the MAP2K layer. One may thereforeexpect to observe results qualitatively similar to whathas been reported here whenever the system has dis-parate timescales regardless of the actual molecular con-centrations and kinetic rates which can vary substantially across different cells [32–34]. Thus, as the MAPK cas-cade is present in all eukaryotic cells [5, 6], the mecha-nism for short-term memory in such a signaling cascadethat is presented here may hold for such cells in general.As the duration of MAPK ∗∗ activity is critical for manycellular decisions [35], e.g., the prolonged activation ofERK resulting in its translocation to the nucleus [36],the persistent reverberatory activity seen here may playa non-trivial role in regulation of cellular functions.The basal level activity of MAPK in a normal cellis maintained at a low proportion of the total MAPKconcentration and serves several biological functions [37].We observe a crossover between two qualitatively distinctregimes of relaxation behavior of MAPK ∗∗ occurring ata steady state that is characterized by relatively low pro-portion of activation of the available MAPK [ ∼ τ P R .It is known that ERK MAPK isoforms (e.g., p42 andp44) are abundantly expressed in non-dividing termi-nally differentiated neurons [27]. Activation of MAPK byspaced stimulation is known to be responsible for mor-phological changes in dendrites [26]. Studies also sug-gest that the activation of the MAPK pathway is linkedwith associative learning in the mammalian nervous sys-tem, synaptic plasticity and neurological memory [25–27, 38, 39]. An intriguing possibility suggested by the re-sults reported here is that the observed repeated spikingin MAPK activity may function as an effective tempo-rally spaced signal to the nucleus of a neuron. This canthen facilitate subsequent changes in the cell required formemory formation.Another well-known example of eukaryotic cellularmemory is observed during chemotactic migration alongthe gradient of a chemical signal [40, 41]. The directional-ity of migration is known to persist for a certain duration,even if the chemical gradient is altered or becomes static.Studies show that the protein Moesin contributes to thelong-lived rigidity of the cytoskeleton assembly that sub-sequently leads to the directional memory in polarizedmigrating cells [41]. However, the intra-cellular processesthat underlie the persistent activity of Moesin in the ab-sence of a gradient mediated signal are still largely un-known. Evidence suggests that the regulation of Moesinand other ERM proteins are linked with the activity ofthe MAPK pathway [42, 43]. The long-term reverber-atory activity of MAPK following the withdrawal of astimulus that is reported here may be a possible mecha-nism underlying such persistent cellular behavior.To conclude, we have shown the possibility of long-lived reverberatory activity in a signaling cascade follow-ing the withdrawal of external stimuli. Our results sug-gest a mechanism through which the intra-cellular sig-naling system can encode short-term memory of signalsto which the cell was previously exposed. The large-amplitude spiking activity of MAPK following the re-moval of a prior stimulus may also provide a mechanismfor signal integration and learning when the cascade isrepeatedly stimulated. We note that there may be addi-tional factors not considered here that may lengthen thepersistence of reverberatory activity, including scaffoldproteins that increase the lifetime of kinase complexes.Our results suggest that the MAPK cascade potentiallyhas a key role in shaping the information processing ca-pabilities of eukaryotic cells in diverse environments.
Acknowledgments
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S1. THE MODEL EQUATIONS
TABLE S1: Components of the MAPK CascadeComponent Notation SymbolMitogen-activated Protein Kinase Kinase Kinase MAP3K 3KSingly Phosphorylated Mitogen-activated Protein Kinase Kinase Kinase MAP3K* 3K*Mitogen-activated Protein Kinase Kinase MAP2K 2KSingly Phosphorylated Mitogen-activated Protein Kinase Kinase MAP2K* 2K*Doubly Phosphorylated Mitogen-activated Protein Kinase Kinase MAP2K** 2K**Mitogen-activated Protein Kinase MAPK KSingly Phosphorylated Mitogen-activated Protein Kinase MAPK* K*Doubly Phosphorylated Mitogen-activated Protein Kinase MAPK** K**MAP3K-Phosphatase 3K P’ase P MAP2K-Phosphatase 2K P’ase P MAPK-Phosphatase K P’ase P K The three layer MAPK cascade comprises the following enzyme-substrate reactions: S + 3 K k −→←−− k − S. K k −→ S + 3 K ∗ P K + 3 K ∗ kp −→←−−− kp − K ∗ .P K kp −→ P K + 3 K K ∗ + 2 K k −→←−− k − K ∗ . K k −→ K ∗ + 2 K ∗ P K + 2 K ∗ kp −→←−−− kp − K ∗ .P K kp −→ P K + 2 K K ∗ + 2 K ∗ k −→←−− k − K ∗ . K ∗ k −→ K ∗ + 2 K ∗∗ P K + 2 K ∗∗ kp −→←−−− kp − K ∗∗ .P K kp −→ P K + 2 K ∗ K ∗∗ + K k −→←−− k − K ∗∗ .K k −→ K ∗∗ + K ∗ P K + K ∗ kp −→←−−− kp − K ∗ .P K kp −→ P K + K K ∗∗ + K ∗ k −→←−− k − K ∗∗ .K ∗ k −→ K ∗∗ + K ∗∗ P K + K ∗∗ kp −→←−−− kp − K ∗∗ .P K kp −→ P K + K ∗ The above enzyme-substrate reactions can be expressed in terms of the following coupled ordinarydifferential equations (ODEs): d [3 K ] dt = k − . [ S. K ] + kp . [3 K ∗ .P K ] − k . [ S ] . [3 K ] ,d [ S. K ] dt = k . [ S ] . [3 K ] − ( k − + k ) . [ S. K ] ,d [3 K ∗ .P K ] dt = kp . [ P f K ] . [3 K ∗ ] − ( kp + kp − ) . [3 K ∗ .P K ] ,d [3 K ∗ ] dt = k . [ S. K ] + kp − . [3 K ∗ .P K ] − kp . [ P f K ] . [3 K ∗ ]+( k − + k ) . [3 K ∗ . K ] − k . [3 K ∗ ] . [2 K ]+( k − + k ) . [3 K ∗ . K ∗ ] − k . [3 K ∗ ] . [2 K ∗ ] ,d [2 K ] dt = k − . [3 K ∗ . K ] + kp . [2 K ∗ .P K ] − k . [3 K ∗ ] . [2 K ] ,d [3 K ∗ . K ] dt = k . [3 K ∗ ] . [2 K ] − ( k − + k ) . [3 K ∗ . K ] ,d [2 K ∗ .P K ] dt = kp . [ P f K ] . [2 K ∗ ] − ( kp + kp − ) . [2 K ∗ .P K ] ,d [2 K ∗ ] dt = k . [3 K ∗ . K ] + kp − . [2 K ∗ .P K ] − kp . [ P f K ] . [2 K ∗ ]+ k − . [3 K ∗ . K ∗ ] − k . [3 K ∗ ] . [2 K ∗ ] + kp . [2 K ∗∗ .P K ] ,d [3 K ∗ . K ∗ ] dt = k . [3 K ∗ ] . [2 K ∗ ] − ( k + k − ) . [3 K ∗ . K ∗ ] ,d [2 K ∗∗ .P K ] dt = kp . [ P f K ] . [2 K ∗∗ ] − ( kp + kp − ) . [2 K ∗∗ .P K ] ,d [2 K ∗∗ ] dt = k . [3 K ∗ . K ∗ ] + kp − . [2 K ∗∗ .P K ] − kp . [ P f K ] . [2 K ∗∗ ]+( k − + k ) . [2 K ∗∗ .K ] − k . [2 K ∗∗ ] . [ K ]+( k − + k ) . [2 K ∗∗ .K ∗ ] − k . [2 K ∗∗ ] . [ K ∗ ] ,d [ K ] dt = k − . [2 K ∗∗ .K ] + kp . [ K ∗ .P K ] − k . [2 K ∗∗ ] . [ K ] ,d [2 K ∗∗ .K ] dt = k . [2 K ∗∗ ] . [ K ] − ( k + k − ) . [2 K ∗∗ .K ] ,d [ K ∗ .P K ] dt = kp . [ P fK ] . [ K ∗ ] − ( kp − + kp ) . [ K ∗ .P K ] ,d [ K ∗ ] dt = k . [2 K ∗∗ .K ] + kp − . [ K ∗ .P K ] − kp . [ P fK ] . [ K ∗ ]+ k − . [2 K ∗∗ .K ∗ ] − k . [2 K ∗∗ ] . [ K ∗ ] + kp . [ K ∗∗ .P K ] ,d [2 K ∗∗ .K ∗ ] dt = k . [2 K ∗∗ ] . [ K ∗ ] − ( k − + k ) . [2 K ∗∗ .K ∗ ] ,d [ K ∗∗ .P K ] dt = kp . [ P fK ] . [ K ∗∗ ] − ( kp − + kp ) . [ K ∗∗ .P K ] ,d [ K ∗∗ ] dt = k . [2 K ∗∗ .K ∗ ] + kp − . [ K ∗∗ .P K ] − kp . [ P fK ] . [ K ∗∗ ] . S ] = [ S ] tot − [ S. K ] , [ P f K ] = [ P K ] − [3 K ∗ .P K ] , [ P f K ] = [ P K ] − [2 K ∗ .P K ] − [2 K ∗∗ .P K ] , [ P fK ] = [ P K ] − [ K ∗ .P K ] − [ K ∗∗ .P K ] . It is explicitly ensured that the total concentrations of all individual kinases and phosphatases are conserved at alltimes. The concentrations of the different molecular species can vary over several orders of magnitudes. We havetherefore numerically solved the equations using low relative and absolute tolerances in order to ensure the accuracyof the resulting time-series.
S2. SYSTEM PARAMETERS
The numerical values for the reaction rates are obtained from Ref. [13], and are listed in Table S2.
TABLE S2: Reaction RatesRate constant Value Units k µM. min) − k −
150 min − k
150 min − kp µM. min) − kp −
150 min − kp
150 min − k µM. min) − k −
30 min − k
30 min − kp µM. min) − kp −
150 min − kp
150 min − k µM. min) − k −
30 min − k
30 min − kp µM. min) − kp −
150 min − kp
150 min − k µM. min) − k −
30 min − k
30 min − kp µM. min) − kp −
150 min − kp
150 min − k µM. min) − k −
150 min − k
150 min − kp µM. min) − kp −
150 min − kp
150 min − TABLE S3: Total concentration (in µM ) of the kinase proteins for Fig. 4 (panels a–c) and Fig. 7[ K ] tot [2 K ] tot [3 K ] tot µM ) of the phosphatase proteins for Figs. 2–6Phosphatase Protein ValueMAP3K-Phosphatase 1 × − MAP2K-Phosphatase 3 × − MAPK-Phosphatase 0.05TABLE S5: Total concentration (in µM ) of the kinase proteins for Figs. 2–3Panels [ K ] tot [2 K ] tot [3 K ] tot (a) and (f) 3.0 3.0 0.0080(b) and (g) 1.0 2.4 0.0024(c) and (h) 1.2 6.0 0.0028(d) and (i) 2.0 2.2 0.0024(e) and (j) 4.8 6.0 0.0014 S3. SUPPLEMENTARY FIGURES
FIG. S1: Magnified views of the phase space trajectory shown in Fig. 2(i). The blue markers correspond to the final point inthe time series displayed in Fig. 2(d), while the red markers indicate the fixed point of the dynamical system in the presence ofstimulus. (a) Magnified view of the trajectory beginning from the black marker shown in Fig. 2(i). The pink marker denotes thestarting point of the segment of the trajectory displayed in panel (b). (b) Further magnification of the section of phase-planetrajectory shown in (a), corresponding to the duration when the system moves away from the unstable limit cycle and convergesto the stable fixed point. FIG. S2: Processes underlying long-lived memory and reverberatory dynamics. (a) Schematic representation of MAPK cascadeshowing the processes that occur subsequent to removing a stimulus. The numbers (1 −
4) represent the sequence of eventsthat lead to the emergence of the post-stimulus large-amplitude spiking activity shown in (b). The enzyme-substrate proteincomplex formed during activation of MAPK by doubly phosphorylated MAP2K is indicated by “c”. The green arrow fromthe MAPK layer to the MAP2K layer represents the release of doubly phosphorylated MAP2K from downstream complexes.(b) A characteristic time-series for the normalized concentration of singly and doubly phosphorylated MAPK ( n K ∗ and n K ∗∗ ,respectively) following the removal of an applied stimulus of amplitude S = 2 . × − µM at t = 0. The numbers (1 − FIG. S3: Characteristic dynamics of the molecular components of the MAP Kinase cascade following withdrawal of a stimulus.(a) The time-series of the normalized concentration of doubly phosphorylated MAPK ([ K ∗∗ ] / [ K ] tot ) following removal of anapplied stimulus with amplitude S = 2 . × − µM at t = 0. (b-r) Time-series of the normalized concentrations of the differentcomponents of the MAPK cascade, shown starting from t = 150 minutes after withdrawing the stimulus, displayed togetherwith the time-series of normalized MAPK activity [ K ∗∗ ] / [ K ] tot . The total concentrations of the kinases and phosphatases usedfor generating the figures are provided in Tables S3 and S4, respectively. FIG. S4: Dependence of reverberatory activity on the total kinase concentrations, viz., MAPK ([ K ] tot ), MAP2K ([2 K ] tot ) andMAP3K ([3 K ] tot ). (a) The number of post-stimulus spikes N r , (b) the total memory time τ m (in minutes), and (c) isosurfacesfor N r observed on withdrawing an applied stimulus of amplitude S = 2 . × − µM , are shown as functions of the three totalkinase concentrations. The total concentrations of the phosphatases are held fixed for (a-c) and are provided in Table S4.FIG. S5: Characterization of the reverberatory dynamics observed after withdrawing a stimulus having amplitude S = 2 . × − µM . (a) The interval between successive spikes i − i increases with time ( t i being the time of occurrence of the i thspike) for two distinct total concentrations of MAP2K. The total concentrations of MAPK and MAP3K are [ K ] tot = 1 . µM and [3 K ] tot = 2 . nM , respectively. (b) The primary recovery time τ PR (stars) and the total duration of reverberatory activity τ r (filled circles) are shown for different values of N r (indicated by the color bar). While τ PR increases monotonically withincreasing total MAPK concentration, τ r shows a more complex dependence ([2 K ] tot = 3 µM and [3 K ] tot = 4 nM ). (c) Thedependence of τ r on [ K ] tot for different values of N r has a similar nature for different choices of [3 K ] tot (indicated by thecolor bar, [2 K ] tot = 3 µM ). Note that for panel (c), we consider only situations where the system attains a steady state onmaintaining the stimulation. For the total concentrations of the phosphatases see Table S4. FIG. S6: Protein complexes in the MAPK cascade exhibit relaxation behavior occurring over a broad range of time-scales.Decay of activity is shown after withdrawing an applied stimulus of amplitude S = 1 . × − µM . The relaxation times τ x of thedifferent molecular species, viz., (a) the protein complexes between non-phosphorylated and singly phosphorylated (non-active)kinase proteins and the doubly phosphorylated (active) kinase protein of the preceding layer, and (b) the protein complexesbetween the phosphorylated (singly- or doubly-) kinase proteins and the phosphatase that carries out dephosphorylation inthe corresponding layer of the MAPK cascade, vary with the total concentration of MAP2K. The nature of this dependenceis distinct for lower and higher values of [2 K ] tot . For both panels, [ K ] tot = 0 . µM and [3 K ] tot = 0 . µM . The totalconcentrations of the phosphatases are provided in Table S4. FIG. S7: Dependence of the primary recovery time τ PR on (a-i) the total concentration of MAP2K ([2 K ] tot ) and on (j-o)the total concentration of MAPK ([ K ] tot ) for different values of the total concentration of MAP3K ([3 K ] tot ), obtained uponremoving stimuli having different amplitudes S . Panels (a,d,g,j,m) are for S = 0 . × − µM , panels (b,e,h,k,n) are for S = 1 . × − µM , and panels (c,f,i,l,o) are for S = 2 . × − µM . We have only considered situations where the systemreaches a steady state upon application of a time-invariant stimulus, and that do not show any reverberatory activity ( N r = 0)during relaxation to the resting state. The curves in panels (a-i) are obtained for different values of [ K ] tot , namely, (a-c)[ K ] tot = 0 . µM , (d-f) [ K ] tot = 1 . µM , and (g-i) [ K ] tot = 1 . µM . The curves in panels (j-o) are obtained for different valuesof [2 K ] tot , namely, (j-l) [2 K ] tot = 0 . µM , and (m-o) [2 K ] tot = 1 . µM . The total concentrations of the phosphatases for allpanels are given in Table S4. FIG. S8: Dependence of reverberatory activity on the total concentrations of the phosphatases MAPK P’ase ([ P K ]), MAP2KP’ase, ([ P K ]) and MAP3K P’ase ([ P K ]). (a) The number of spikes N r and (b) the total memory time τ m (in minutes)observed on withdrawing an applied stimulus of amplitude S = 2 . × − µM . Situations where the primary recovery time islonger than a maximum or cut-off value (see Methods), such that the reverberatory nature of the dynamics cannot be properlymeasured, are indicated by the color corresponding to “U”. (c) The interval between successive spikes i − i increaseswith time ( t i being the time of occurrence of the i th spike). As the MAPK P’ase concentration is increased, the durations ofthese intervals are seen to increase. The total concentrations of the other two phosphatases are maintained at [ P K ] = 680 pM and [ P K ] = 10 pM . (d) The variation of primary recovery time τ PR (stars) and the total duration of reverberatory activity τ r (filled circles) as a function of total MAPK P’ase concentration are shown for different values of N r (indicated by the colorbar). While τ PR decreases monotonically with increasing [ P K ], τ r shows a more complex dependence ([ P K ] = 200 pM and[ P K ] = 6 pM ). (e) Dependence of the total memory time τ m on total MAPK P’ase concentration ([ P K ] shown in log scale) fordifferent total concentrations of MAP2K P’ase (values indicated above each of the three panels) and MAP3K P’ase (indicatedusing different colors as shown in the color bar). Note that we consider only situations where the system attains a steady stateon maintaining stimulation. For details of the total concentrations of the kinases, see Table S3. FIG. S9: The time interval between successive spikes i − i obtained after removing a stimulus, increases with the numberof spike events ( i being the event number of the i th spike). The trend appears to be independent of the stimulus amplitude S . The total concentrations of the phosphatases are P K = 0 . µM , P K = 680 pM and P K = 10 pMpM