Non-associative learning in intra-cellular signaling networks
aa r X i v : . [ q - b i o . S C ] J u l Non-associative learning in intra-cellular signaling networks
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 400094, India. (Dated: July 4, 2018)Nonlinear systems driven by recurrent signals are known to exhibit complex dynamical responseswhich, in the physiological context, can have important functional consequences. One of the simplestbiological systems that is exposed to such repeated stimuli is the intra-cellular signaling network. Inthis paper we investigate the periodic activation of an evolutionarily conserved motif of this network,viz., the mitogen-activated protein kinase (MAPK) signaling cascade, with a train of pulses. Theresulting response of the cascade, which shows integrative capability over several successive pulses,is characterized by complex adaptive behavior. These include aspects of non-associative learning, inparticular, habituation and sensitization, which are observed in response to high- and low-frequencystimulation, respectively. In addition, the existence of a response threshold of the cascade, anapparent refractory behavior following stimulation with short inter-pulse interval, and an alternans-like response under certain conditions suggest an analogy with excitable media.
Nonlinear systems can respond to variations in theirenvironment by exhibiting a wide range of complex dy-namical patterns [1–6] that may often be functionallysignificant [7–13]. These variations are commonly asso-ciated with natural cycles such as the diurnal rhythm.In particular, biological systems are typically subjectedto periodic stimuli with frequencies that can vary over awide range of time scales, viz., from ultradian to infra-dian rhythms [14–16]. Examples include the entrainmentof the circadian clock to the day-night cycle [17], varia-tions in hormonal levels over a period of a month thatdrive the menstrual cycle [18] and calcium oscillations atthe time-scale of minutes modulating the efficiency andspecificity of gene expression [19]. Of all the biologicalsystems capable of exhibiting complex functionally sig-nificant responses when driven by periodic stimuli, per-haps one of the simplest is the intra-cellular signalingnetwork [20]. In its natural environment, the membrane-bound receptors of a cell may repeatedly be stimulatedon encountering ligands, for instance as a consequenceof pulsatile variations in hormones [21]. Cellular func-tions may also be modulated by internal cues that varyperiodically, e.g., oscillations in the concentrations of in-tracellular messengers such as Ca [22, 23] and cyclicAMP [24, 25]. It is therefore important to investigatehow key components of the signaling network in the cellrespond when subjected to repeated stimuli in the formof pulse trains.One of the most ubiquitous motifs of this network isthe mitogen-activated protein kinase (MAPK) cascade,which is found across all eukaryotic cells [26, 27]. It con-sists of a sequential arrangement of three types of pro-tein kinase, viz., MAPK, MAPK kinase (MAP2K) andMAPK kinase kinase (MAP3K). The activated kinase ineach layer of the cascade functions as an enzyme for phos-phorylating (and thus activating) the kinase in the layerimmediately downstream. The subsequent deactivationis mediated by the corresponding dephosphorylating en-zyme known as phosphatases (P’ase). Despite its struc-tural simplicity this motif is involved in regulating a wide array of vital cellular functions, including proliferationand apoptosis [27], stress response [28] and gene expres-sion [29]. Activation of the cascade is initiated when ex-tracellular ligands stimulate membrane-bound receptors,or when intracellular cues occur upstream of the cascade,with the information being relayed to MAP3K through aseries of intermediaries. The terminal kinase of the mo-tif (MAPK), transmits the signal further downstream byphosphorylating various proteins including transcriptionregulators [30]. The behavior of the cascade subjected tosustained stimulation has been extensively investigated,and the existence of several emergent features has beenobserved. These include ultrasensitivity [31], bistabilitywhich allows the system to switch between two statescorresponding to low and high activity [32–36] and os-cillations [36–41]. In earlier work, we have shown thatthe cascade stimulated with a pulse of finite duration re-sponds with a rich variety of transient behavior, includingphenomena indicative of the presence of short-term mem-ory [42]. The complex modulations seen in the responseof the cascade are crucially dependent on the interactionsbetween the time-scales of the intrinsic processes and thatof the applied stimulus. It is thus intriguing to considerhow the system will respond to repeated stimulation.In this paper, we investigate the dynamics of theMAPK cascade that is stimulated by periodic trains ofpulses. Despite the absence of any explicit feedback,under suitable conditions we find that the system dis-plays adaptive behavior including non-associative learn-ing [43, 44], viz., habituation (desensitization) and sensi-tization. These allow plasticity in the behavioral reper-toire of the intracellular signaling motif by enabling mod-ification of the strength, duration and even the qualita-tive nature of its response to recurrent stimulation. Inaddition to these, we report the occurrence of a tem-poral sequence of strong and weak responses to succes-sive pulses reminiscent of the phenomenon of “alternans”in excitable cells [45, 46]. This, coupled with the exis-tence of a response threshold and an apparent refractorybehavior when subjected to high-frequency stimulationstrongly suggests an analogy with excitable media [47].While learning is commonly associated with the behav-ior at the level of organisms [48–53], it is intriguing thatrudimentary forms of such complex adaptive responsescan be seen in a simple network of sub-cellular compo-nents. As the MAPK signaling cascade is involved in co-ordinating diverse processes in all eukaryotic cells, theseresults point to the potential functional utility of suchemergent dynamical phenomena in biological systems.We have simulated the dynamics of the three layer ki-nase cascade using the Huang-Ferrell model of the MAPKsignaling motif, schematically illustrated in Fig. 1 (a).This model consists of 10 enzyme-substrate reactions de-scribed by 18 coupled differential equations [31]. Eachof the several kinase and phosphatase-mediated enzyme-substrate reactions in the cascade consist of (i) a re-versible enzyme-substrate complex formation step, and(ii) an irreversible step corresponding to the activa-tion/deactivation of a kinase (see Supplementary Infor-mation for details). The ratio of the activation and de-activation rates range over four orders of magnitude [36],underlining the vast diversity of dynamical time-scalespresent in the system. The equations are numericallysolved without invoking the quasi-steady-state hypothe-sis [54]. We explicitly ensure that the total concentra-tions of each of the constituent kinases in the system areconserved. In our simulations, we assume that the cas-cade is initially in the resting state, where the kinases arecompletely non-phosphorylated. Following the exposureof the cascade to a train of pulses, we record the resultingresponse pattern, viz., the MAPK activity.Investigations into the dynamics of the Huang-Ferrellmodel [31] have typically focused on the asymptotic re-sponse of the cascade to sustained stimulation. In con-trast, here we investigate the response of the systemwhen it is subjected to recurrent activation by periodicstimuli. Specifically, we consider a signal comprising atrain of pulses, each having amplitude S , duration P and separated from each other by an inter-pulse inter-val I [Fig. 1 (a)]. The cascade is released from stim-ulation between two successive pulses, and attempts torelax back to its resting state. On arrival of the nextpulse, the cascade is activated once more, albeit beforeit has completely relaxed. This, coupled with the mul-tiple time-scales of activation and relaxation present inthe system, results in non-trivial adaptive temporal re-sponse. Selected examples of such behavior are shown inFig. 1 (b-d). These different time series of the activatedMAPK concentration (normalized with respect to the to-tal MAPK concentration) correspond to the cascade be-ing subjected to pulse trains characterized by differentparameter values of P and I .Fig. 1 (b) displays the response of the system sub-jected to high-frequency stimulation by short-durationpulses. Here, starting from its resting state value, eachsubsequent pulse elicits a slightly higher response of n ∗∗ K until the peak activation suddenly spikes to a value closeto its saturation. This behavior can be interpreted as a FIG. 1: Non-associative learning in a MAPK cascade stim-ulated by a pulse train. (a) Schematic representation of alinear three-layer MAPK cascade whose component kinasesare activated/deactivated by the addition/removal of phos-phate groups through phosphorylation/dephosphorylation re-spectively. Signaling is initiated when MAPK kinase kinase(MAP3K) is activated by a periodic signal comprising a seriesof pulses having amplitude S and duration P , separated byinter-pulse interval I . For the cases investigated here, the cas-cade receives no stimulus between two successive pulses. Theresponse of the cascade to the signal is measured in termsof MAPK activity, viz., the normalized concentration n ∗∗ K ofdoubly phosphorylated MAPK. (b-d) Time series represent-ing qualitatively different adaptive responses of the cascadeto pulse trains characterized by a range of S , P and I . Theshaded bars correspond to the intervals during which MAP3Kis stimulated. (b) Desensitization behavior of the cascade cor-responding to an attenuated response on persistent exposureto the periodic stimulus. (c) Sensitization of the cascade char-acterized by a low level of MAPK activity on initial exposurefollowed by stronger responses upon repeated stimulation. (d)Alternating high and low levels of MAPK activity (“alter-nans”) in response to successive pulses. (e) Threshold-likeresponse to the pulse duration P of the maximum MAPK ac-tivity for a fixed set of values of the signal strength S andinter-pulse duration I of the pulse train. (f) Nonlinear de-pendence of the MAPK cascade response on the inter-pulseinterval for a pair of pulses (shaded bars). For details of sys-tem and signal parameter values used see SI. form of signal integration, and may be repeated multipletimes as the pulse train is continued. However, for anappropriate range of P and I (as in the figure), after agiven number of pulses we observe behavior analogous to desensitization when the system no longer shows spikingactivity even for sustained periodic stimulation. Thus,following an initial large amplitude response, the subse-quent activity of the system is attenuated even thoughthe nature of the received signal remained unchanged.When the cascade is stimulated instead by low-frequencypulse trains having relatively longer pulse durations, weobserve a phenomenon analogous to sensitization . Herethe cascade exhibits low-level activity on receiving theinitial pulse but switches to high-amplitude spiking inresponse to all subsequent pulses [Fig. 1 (c)]. Thus, theinitial low-level activity effectively “primes” the cascadeto reach response levels close to saturation. This occursbecause of the existence of long relaxation time-scales incertain components of the cascade, allowing for responseaccumulation over successive stimulations. Decreasing P by a small amount gives rise to a qualitatively distinctphenomenon characterized by alternating low and highpeaks of MAPK activity, reminiscent of alternans [46].As alternans is a phenomenon that is associated withexcitable media, it is intriguing to consider whether theperiodically stimulated cascade exhibits other character-istics of such systems, in particular, the existence of aresponse threshold [47]. As seen in Fig. 1 (e), there isindeed a large discontinuous change in the peak activa-tion n mK ∗∗ of MAPK when the pulse duration P crosses aspecific value P c that depends on the choice of S and I .Extending the analogy with excitable media, we find thatthe cascade also exhibits a nonlinear relation between itsresponse to successive pulses and the inter-pulse inter-val. This can be seen from the behavior displayed inFig. 1 (f) where the cascade is stimulated by a pair ofpulses separated by an interval I . When I is reduced,the response duration resulting from the second pulseincreases in comparison to the duration of the responsecaused by the first. As an aside, we note that for theparameter regime considered here, the system exhibitspost-stimulus reverberatory activity [42].Fig. 2 (a-f) depicts representative time-series showingthe activity of the cascade on either side of the responsethreshold, obtained for different choices of the periodicstimulation parameters. We note that in all of the casesshown here, the system shows a gradual build-up of activ-ity over multiple pulses before reaching asymptotic peakactivity levels. This corresponds to signal integration(mentioned earlier) where the response of the system tosuccessive stimuli is modulated by the preceding stim-uli. Fig. 2 (a) shows a typical subthreshold response( sub ), where the peak MAPK activity is highly atten-uated ( <
5% of the saturation response value). Notethat the nature of the response (i.e., whether it is sub-or supra-threshold) is a function of all three stimulationparameters S , P and I . For instance, for the same sig-nal strength S considered in panel (a), the steady-state FIG. 2: (a-f) Characteristic responses of the MAPK cas-cade to stimulation of MAP3K by a train of pulses, eachof amplitude S having duration P , with inter-pulse inter-val I : (a) attenuated response of the cascade characterizedby sub-threshold activity (sub), (b) large-amplitude spikingresponses characterizing supra-threshold activity (sup), (c)prolongation of supra-threshold activity duration (pro) on ap-plication of a signal having pulses with longer duration, (d)coexistence (cox) of sub- and supra-threshold activity which,for a range of I , results from the integration of responses overseveral preceding pulses, (e) desensitization (des), where in-tegration over multiple successive pulses results in a supra-threshold spiking response but subsequently only exhibitssub-threshold activity, and (f) sensitization (sen), where sub-threshold activity in response to the initial pulse gives way tosupra-threshold activity for all subsequent pulses. The shadedbars correspond to the intervals during which MAP3K is stim-ulated. (g) Dependence of the cascade response on the pulsestrength S and duration P for three different values of theinter-pulse interval I . The colors represent the nature of theresponse [classified into the categories (a-f) mentioned above].(h-i) Magnified views of the P − S planes for (h) I = 105 and(i) 2000 minutes show the regions corresponding to desensi-tization and sensitization, respectively. (j) The variation ofthe critical value of pulse duration P c , above which the cas-cade exhibits supra-threshold response, with pulse amplitude S . The curves correspond to pulse trains having differentinter-pulse intervals I (as shown in the colorbar). response of the cascade would have been close to satu-ration if the stimulation had been applied in a sustainedfashion (i.e., I → sup , Fig. 2 (b)]. On varying the different stimula-tion parameters we observe other types of suprathresholdactivity. For example, on increasing P alone (or alterna-tively, S alone), the system exhibits prolongation of thepeak activity close to saturation [ pro , Fig. 2 (c)]. Forhigh-frequency stimulation (i.e., low I ) after a transientperiod we observe suprathreshold peak responses only af-ter every N pulses for values of P and S that lie betweenthose giving rise to sub and sup responses [see the low-est plane of Fig. 2 (g)]. This response behavior, whichcorresponds to the coexistence ( cox ) of peak activity lev-els having different amplitudes (ranging from values justabove zero to near-saturation) is shown in Fig. 2 (d). Forlower frequency stimuli, the cox regime corresponds to M : 1 response where multiple peaks in MAPK activity,whose amplitudes can again vary widely, are observed inresponse to each pulse [not shown]. Apart from these, wealso observe behavior corresponding to non-associativelearning, viz., desensitization [ des , Fig. 2 (e)] and sensiti-zation [ sen , Fig. 2 (f)], as described earlier. Specifically,at the interface of the cox and sub regions in the stimula-tion parameter space, the des response regime is observedfor high-frequency pulse trains [Fig. 2 (h)] while for low-frequency stimulation we obtain sen [Fig. 2 (i)]. We notethat for higher values of S , the transition from cox to sub gets sharper thereby reducing the range of P over whichthe des and sen phenomena are observed. An overviewof the stimulation parameter space is given in Fig. 2 (g)indicating the conditions for which each of the responsesdescribed above can be obtained.The “learning” behavior associated with the period-ically stimulated cascade is seen in the vicinity of theresponse threshold mentioned earlier corresponding tothe boundary of the sub regime [Fig. 2 (g)]. Hence, weexamine the dependence of the threshold on the stimu-lation parameters in Fig. 2 (j). The reciprocal relationbetween the signal strength S and the critical pulse dura-tion P c necessary for suprathreshold response seen overa wide range of S suggests that the total signal inten-sity of a pulse, measured as the product of S and P ,determines the threshold. Deviation from this simplerelation is observed for sufficiently low signal strength.This implies that a minimal value of S is required to ob-serve suprathreshold response, regardless of the durationfor which the pulse is maintained. We note that in thelimit of I →
0, this minimal signal strength correspondsto the lower critical value required to observe a transi-tion from low level of MAPK activity to high-amplitudeoscillations when the cascade is subjected to sustainedstimulation [36, 42]. As I is increased, we observe thatthe response threshold (measured in terms of the criticalpulse duration P c ) increases, which suggests that the ex-citability of the system reduces as the frequency of the periodic stimulus decreases.The phenomena reported here are robust with re-spect to variations in the model parameters around thevalues used in this paper, including the kinetic rateconstants and the molecular concentrations of the con-stituent kinases and phosphatases. We have also ob-served similar behavior with cascades having branchedarchitecture, e.g., MAP3K activating two different typesof MAP2K [55]. 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.A mechanistic understanding of the phenomena re-ported here is made difficult by the large number ofcoupled dynamical variables in the model that operateacross different time-scales. This complexity may be un-tangled by using the framework of excitable systems. Asalluded to earlier, many of the characteristic features as-sociated with excitability are present in the system inves-tigated here. These include the existence of two qualita-tively distinct states of activation separated by a thresh-old [Fig. 1 (e)], nonlinear response to repeated stimu-lation [Fig. 1 (f)], an apparently refractory behavior asseen most prominently during desensitization [Fig. 1 (b)]and phenomena analogous to alternans [Fig. 1 (d)]. Thisappealing analogy provides a means by which a phe-nomenological understanding of the emergent behaviorof this complex system might be achieved. We note thatthe excitability paradigm has been invoked earlier to ex-plain aspects of cellular activity in the context of antigenrecognition by T cells [56, 57]. Our results show that theemergent dynamics of MAPK cascade, which is knownto mediate immune response [58], provides an explicitmechanistic basis for such a theoretical framework forexplaining the adaptive response of the immune systemto its microenvironment.Among the functionally significant dynamical phenom-ena reported here, “learning” is perhaps the most in-triguing. It confers on the system the ability to modifyits behavior in response to information, which is criticalfor adapting to a changing environment. The capabil-ity to learn often presupposes the existence of a feed-back that allows bidirectional communication betweenthe components associated with receiving a signal andthose that initiate a corresponding response [59]. In thekinase cascade investigated here, an explicit feedback isabsent as each layer activates the one immediately down-stream. However, an implicit feedback results from theinherent features of kinase activation, viz., sequestrationand multi-site phosphorylation [33, 34, 36, 55]. This canhave non-trivial consequences, such as the appearanceof short-term memory, even when the MAPK cascade issubjected to a single pulse [42].To conclude, in this paper we have shown that a richrepertoire of responses can be obtained when the systemis exposed to a train of pulses. This results from the im-plicit feedback, which orchestrates an interplay betweenthe periodic stimulus and the diverse activation and re-laxation time-scales of the signaling components. In par-ticular, the system can exhibit sensitization and desensi-tization, which are examples of non-associative learning.These may play an important role in the cell’s ability tofunction in its natural environment, where it is continu-ally exposed to signals of varying intensity and duration.This necessitates an ability to respond selectively to thereceived stimuli. Such adaptive mechanisms allow the cellto ignore persistent background stimuli through habitu-ation (desensitization) but respond strongly to signals towhich it has been primed through earlier exposure (sensi-tization). Given that a single linear cascade exhibits suchcomplex adaptive behavior, it is intriguing to speculateabout the potential capabilities inherent in the coordi- nated action of multiple subcellular processes [60]. Themechanism through which learning at the sub-cellularscale can impact adaptive behavior in an organism atcellular and possibly higher scales remains an intriguingquestion.SNM is supported by the IMSc Complex SystemsProject (12 th Plan). The simulations required for thiswork were done in the High Performance Computing fa-cility (Nandadevi and Satpura) of The Institute of Math-ematical Sciences, which is partially funded by DST(Grant no. SR/NM/NS-44/2009). We thank James Fer-rell, Upinder Bhalla, Tharmaraj Jesan, Uddipan Sarma,Bhaskar Saha, Jose Faro, Vineeta Bal, J. Krishnan,Mukund Thattai, Marsha Rosner and Pamela Silver forhelpful discussions. [1] J. Testa, J. P´erez, and C. Jeffries, Phys. Rev. Lett. ,714 (1982) doi:10.1103/PhysRevLett.48.714[2] L. Glass, A. L. Goldberger, M. Courtemanche, andA. Shrier, Proc. R. Soc. Lond. A , 9 (1987).doi:10.1098/rspa.1987.0097[3] M. C. Cross and P. C. Hohenberg, Rev. Mod. Phys. ,851 (1993). doi:10.1103/RevModPhys.65.851[4] A. L. 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SUPPLEMENTARY INFORMATIONfor“Non-associative learning in intra-cellular signaling networks”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 400094, India.
I. 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 ∗∗ ] . where [ 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.0
II. SYSTEM PARAMETERS
The numerical values for the reaction rates used in all our simulations are obtained from Ref. [55], and are listedin Table S2. Please note that these values of kinetic rate constants are very close to that of Huang-Ferrell basevalues [31].
TABLE S2: Reaction RatesRate constant Our base value Huang-Ferrell value Units k µM. min) − k −
150 150 min − k
150 150 min − kp µM. min) − kp −
150 150 min − kp
150 150 min − k µM. min) − k −
30 150 min − k
30 150 min − kp µM. min) − kp −
150 150 min − kp
150 150 min − k µM. min) − k −
30 150 min − k
30 150 min − kp µM. min) − kp −
150 150 min − kp
150 150 min − k µM. min) − k −
30 150 min − k
30 150 min − kp µM. min) − kp −
150 150 min − kp
150 150 min − k µM. min) − k −
150 150 min − k
150 150 min − kp µM. min) − kp −
150 150 min − kp
150 150 min − TABLE S3: Signal parameters for the panels in Fig. 1Parameter (b) (c) (d) (e) (f) Units S × − µMP
71 372 371 60 – 80 300 minsI mins
TABLE S4: Signal parameters for the panels in Fig. 2Parameter (a) (b) (c) (d) (e) (d) Units S × − µMP
49 100 300 50 49.5 287 minsI
105 105 105 105 105 2000 mins
TABLE S5: Total concentration (in µM ) of the kinases and phosphatase proteins for Figs. 1–2Protein Value[ K ] tot K ] tot K ] tot × − MAP2K-Phosphatase 3 × −4