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- Journal List
- J Biol Phys
- v.35(4); Oct 2009
- PMC2750743

# Glutamate regulation of calcium and IP_{3} oscillating and pulsating dynamics in astrocytes

^{1}Mati Goldberg,

^{1}Vladislav Volman,

^{2,}

^{3}Hugues Berry,

^{4}and Eshel Ben-Jacob

^{}

^{1,}

^{2}

^{1}School of Physics and Astronomy, Tel Aviv University, 69978 Ramat Aviv, Israel

^{2}Center for Theoretical Biological Physics, UCSD, La Jolla, CA 92093-0319 USA

^{3}Computational Neurobiology Lab, The Salk Institute, La Jolla, CA 92037 USA

^{4}Project-Team Alchemy, INRIA Saclay, 91893 Orsay, France

^{}Corresponding author.

**This article has been corrected.**See J Biol Phys. 2009 December 19; 36(2): 221.

## Abstract

Recent years have witnessed an increasing interest in neuron–glia communication. This interest stems from the realization that glia participate in cognitive functions and information processing and are involved in many brain disorders and neurodegenerative diseases. An important process in neuron–glia communications is astrocyte encoding of synaptic information transfer—the modulation of intracellular calcium (Ca^{2+}) dynamics in astrocytes in response to synaptic activity. Here, we derive and investigate a concise mathematical model for glutamate-induced astrocytic intracellular Ca^{2+} dynamics that captures the essential biochemical features of the regulatory pathway of inositol 1,4,5-trisphosphate (IP_{3}). Starting from the well-known two-variable (intracellular Ca^{2+} and inactive IP_{3} receptors) Li–Rinzel model for calcium-induced calcium release, we incorporate the regulation of IP_{3} production and phosphorylation. Doing so, we extend it to a three-variable model (which we refer to as the *ChI* model) that could account for Ca^{2+} oscillations with endogenous IP_{3} metabolism. This *ChI* model is then further extended into the G-*ChI* model to include regulation of IP_{3} production by external glutamate signals. Compared with previous similar models, our three-variable models include a more realistic description of IP_{3} production and degradation pathways, lumping together their essential nonlinearities within a concise formulation. Using bifurcation analysis and time simulations, we demonstrate the existence of new putative dynamical features. The cross-couplings between IP_{3} and Ca^{2+} pathways endow the system with self-consistent oscillatory properties and favor mixed frequency–amplitude encoding modes over pure amplitude–modulation ones. These and additional results of our model are in general agreement with available experimental data and may have important implications for the role of astrocytes in the synaptic transfer of information.

**Electronic supplementary material** The online version of this article (doi:10.1007/s10867-009-9155-y) contains supplementary material, which is available to authorized users.

**Keywords:**Inositol 1,4,5-trisphosphate metabolism; Calcium signaling; Pulsating dynamics; Information encoding; Phase locking

## Introduction

Astrocytes, the main type of glial cells in the brain, do not generate action potentials as neurons do, yet they can transfer information to other cells and encode information in response to external stimuli by employing “excitable”-like rich calcium (Ca^{2+}) dynamics [1]. Recognition of the potential importance of the intricate inter- and intracellular astrocyte dynamics has motivated, in recent years, intensive experimental efforts to investigate neuron–glia communication. Consequently, it was discovered that intracellular Ca^{2+} levels in astrocytes can be regulated by synaptic activity [2–6]. Responses to low-intensity synaptic stimulation or spontaneous astrocyte activity usually consist of spatially confined Ca^{2+} transients [3, 4, 7]. On the other hand, high-intensity synaptic activity or stimulation of adjacent sites within the same astrocytic process are generally associated with Ca^{2+} oscillations [8] that can bring forth propagation of both intracellular and intercellular waves [9–11]. Concomitantly, elevation of cytoplasmic Ca^{2+} induces the release from astrocytes of several neurotransmitters (or “gliotransmitters”), including glutamate, ATP, or adenosine (see Evanko et al. [12] for a review). These astrocyte-released gliotransmitters feed back onto pre- and postsynaptic terminals. This implies that astrocytes regulate synaptic information transfer [13–15]. Astrocytes can also mediate between neuronal activity and blood circulation [16], thus extending neuron–astrocyte communications to the level of neuronal metabolism [17].

The physiological meaning of astrocytic Ca^{2+} signaling remains currently unclear, and a long-standing question is how it participates in the encoding of synaptic information transfer [1, 18, 19]. Some of the available experimental data suggest a preferential frequency modulation (FM) mode of encoding, namely synaptic activity would be encoded in the frequency of astrocytic Ca^{2+} pulsations [20]. Indeed, cytoplasmic Ca^{2+} waves in astrocytes often appear as pulse-like propagating waveforms (namely pulses of width much smaller than their wavelength), whose frequency increases when the frequency or the intensity of synaptic stimulation grows [3].

Notwithstanding, the possibility of amplitude modulation (AM) encoding of synaptic activity or even of mixed amplitude and frequency modulation (AFM) encoding has also consistently been inferred [21]. For instance, the amplitude of Ca^{2+} oscillations in response to external stimuli can be highly variable, depending on the intensity of stimulation [2, 11, 22]. Experimental evidence suggests that Ca^{2+} dynamics does not simply mirror synaptic activity but is actually much more complex, to a point that astrocytes are suspected of genuine synaptic information processing [23]. The emerging picture is that the properties of Ca^{2+} oscillations triggered by neuronal inputs in astrocytes (including their amplitude, frequency, and propagation) are likely to be governed by intrinsic properties of both neuronal inputs and astrocytes [1, 3].

From the modeling point of view, simplified or two-variable models for intracellular Ca^{2+} signaling can, in principle, be used to account for the diversity of the observed Ca^{2+} dynamics when the biophysical parameters are varied. We recently presented evidence that one of these two-variable models proposed by Li and Rinzel [24] actually predicts that the same cell could encode information about external stimuli by employing different encoding modes. In this model, changes of biophysical parameters of the cell can switch among AM of Ca^{2+} oscillations, FM of Ca^{2+} pulsations, or combined AM and FM (AFM) Ca^{2+} pulsations [18, 19]. We emphasize that one of the cardinal simplifications of the Li–Rinzel model is neglecting the regulation of inositol 1,4,5-trisphosphate (IP_{3}) dynamics, that is its production and degradation. Since IP_{3} production is regulated by synaptic activity (via extracellular glutamate signaling), IP_{3} dynamics has to be included for proper modeling of synapse–astrocyte communication. Only such modeling can provide a realistic account of astrocytic Ca^{2+} variations induced by nearby synaptic inputs.

Here, we introduce and investigate a concise model for glutamate-induced intracellular astrocytic dynamics. Using this model, we show new putative features of Ca^{2+} dynamics that can have important implications for the role of astrocytes in synaptic information transfer. Our model incorporates current biological knowledge related to the signaling pathways leading from extracellular glutamate to intracellular Ca^{2+}, via IP_{3} regulation and IP_{3}-dependent Ca^{2+}-induced Ca^{2+} release (CICR). First, we extend the Li–Rinzel model to incorporate the regulation of IP_{3}. This yields a three-variable model, called hereafter the “*ChI*” model, for its state variables that are the intracellular Ca^{2+} level *C*, the fraction of inactive IP_{3} receptors *h*, and the available IP_{3} concentration *I*. Similar three-variable models have already been introduced in previous works [25–30] (see Falcke [31] for a review), yet our modeling includes a more realistic description of IP_{3} dynamics, in particular with regard to the complex regulatory pathways of IP_{3} formation and degradation. Furthermore, while we reduce these complex regulatory pathways to a concise mathematical description, we make sure to keep their essential nonlinearities. We then model the contribution of glutamate signals to IP_{3} production and include this contribution as an additional production term into the IP_{3} equation of the *ChI* model. We refer to this case as the “G-*ChI*” model.

We utilize bifurcation theory to study the coexistence of various encoding modes of synaptic activity by astrocytes: AM, pulsation FM, and mixed AFM. We also present results of time simulations of the model, illustrating the richness of intracellular Ca^{2+} dynamics (hence, of the encoding modes) in response to complex time-dependent glutamate signals.

We note that although the model presented here is derived for the specific case of astrocytes, our approach can be readily adopted to model Ca^{2+} dynamics in other cell types whose coordinated activity is based on intra- and intercellular Ca^{2+} signaling, such as heart cells, pancreas cells, and liver cells.

## Derivation of the three-variable *ChI* model of intracellular Ca^{2+} dynamics

In this section, we describe the concise *ChI* model for intracellular Ca^{2+} dynamics in astrocytes with realistic IP_{3} regulation. Given the relative intricacy of this signaling pathway (see Fig. Fig.1),1), each basic building block of the model is described separately in the next sections.

**a**production and

**b**degradation of inositol 1,4,5-trisphosphate (IP

_{3}), summarize the complexity of the signaling network underlying glutamate-induced intracellular dynamics of this second messenger. A peculiar feature of IP

_{3}metabolism

**...**

### CICR core and the two-variable Li–Rinzel model

Intracellular Ca^{2+} levels in astrocytes (as in most other cell types) can be modulated by several mechanisms. These include Ca^{2+} influx from the extracellular space or controlled release from intracellular Ca^{2+} stores such as the endoplasmic reticulum (ER) and mitochondria [33]. In astrocytes, though, IP_{3}-dependent CICR from the ER is considered the primary mechanism responsible of intracellular Ca^{2+} dynamics [34].

Calcium-induced Ca^{2+} release (see Fig. Fig.1c)1c) is essentially controlled by the interplay of two specific transports: efflux from the ER to the cytoplasm that is mediated by Ca^{2+}-dependent opening of the IP_{3} receptor (IP_{3}R) channels and influx into the ER which is due to the action of sarco-endoplasmic reticulum Ca^{2+}-ATPase (SERCA) pumps. In basal conditions, however (when CICR is negligible), intracellular Ca^{2+} levels are set by the respective contributions of a passive Ca^{2+} leak from the ER, SERCA uptake, and plasma membrane Ca^{2+} transport [35, 36].

When synaptic activity is large enough, synaptically released glutamate may spill over the synaptic cleft and bind to the extracellular part of astrocytic metabotropic glutamate receptors (mGluRs) [4]. Binding of glutamate to mGluRs increases cytosolic IP_{3} concentration and promotes the opening of a few IP_{3}R channels [37]. As a consequence, intracellular Ca^{2+} slightly increases. Since the opening probability of IP_{3}R channels nonlinearly increases with Ca^{2+} concentration [38], such an initial Ca^{2+} surge increases the opening probability of neighboring channels. In turn, this leads to a further increase of cytoplasmic Ca^{2+}. These elements therefore provide a self-amplifying release mechanism (hence the denomination of CICR). The autocatalytic action of Ca^{2+} release, however, reverses at high cytoplasmic Ca^{2+} concentrations, when inactivation of IP_{3}R channels takes place, leading to CICR termination [39]. In parallel, SERCA pumps, whose activity increases with cytoplasmic Ca^{2+} [40], quickly sequester excess cytoplasmic Ca^{2+} by pumping it back into the ER lumen. The intracellular Ca^{2+} concentration consequently recovers toward basal values which suppress IP_{3}R channel inactivation. Hence, if glutamatergic stimulation is prolonged, intracellular IP_{3} remains high enough to repeat the cycle, and oscillations are observed [41].

The SERCA pump rate can be taken as an instantaneous function of cytoplasmic [Ca^{2+}] (denoted hereafter by *C*) by assuming a Hill rate expression with exponent 2 (see Appendix 1):

where *v*_{ER} is the maximal rate of Ca^{2+} uptake by the pump and *K*_{ER} is the SERCA Ca^{2+} affinity, that is the Ca^{2+} concentration at which the pump operates at half of its maximal capacity [42].

The nonspecific Ca^{2+} leak current is assumed to be proportional to the Ca^{2+} gradient across the ER membrane by *r*_{L}, the maximal rate of Ca^{2+} leakage from the ER:

where *C*_{ER} is the Ca^{2+} concentration inside the ER stores [36].

IP_{3}R channels can be thought of as ensembles of four independent subunits with three binding sites each: one for IP_{3} and two for Ca^{2+}. The latter sites include an activation site and a separate site for inactivation [36]. IP_{3}-binding sensitizes the receptor toward activation by Ca^{2+} but only if both IP_{3} and activating Ca^{2+}, are bound to a fixed set of three out of four subunits, the channel is open.

Assuming that the kinetic rates of the binding reactions are ordered such as IP_{3}-binding >> Ca^{2+}-activation >> Ca^{2+}-inactivation, Li and Rinzel proposed the following equation for the Ca^{2+} current through the IP_{3}R channels [24]:

with the channel open probability given by , where *m*_{∞}=Hill( *I*,*d*_{1} ), , and *h* account for the three gating reactions, respectively, IP_{3}-binding, activating Ca^{2+}-binding, and Ca^{2+}-dependent inactivation of the receptor. The power of 3 was directly suggested by experimental data [36, 38]. Finally, *I* stands for the intracellular IP_{3} concentration and *r*_{C} is the maximum channel permeability.

Since Ca^{2+} fluxes across the plasma membrane have been proven not necessary for the onset of CICR [35, 43, 44], they can be neglected, so that the cell-averaged total free Ca^{2+} concentration (*C*_{0}) is conserved. Hence, the ER Ca^{2+} concentration (*C*_{ER}) can be rewritten in terms of equivalent cell parameters as *C*_{ER}=(*C*_{0}−*C*)/*c*_{1} where *c*_{1} is the ratio between the ER and the cytosol volumes. It follows that *J*_{chan} and *J*_{leak} can entirely be expressed as functions of cell parameters, namely:

Adding together the above terms (1) and (4), the cytoplasmic Ca^{2+} balance is given by:

This equation is coupled with an equation for *h* that accounts for the kinetics of IP_{3}Rs [24]:

where:

Equations 5 and 6 form the so-called Li–Rinzel (L–R) model of CICR and constitute the core mechanism of our model for astrocyte Ca^{2+} signaling. We discuss below some of its properties.

### AM, FM, and AFM encoding modes in the Li–Rinzel model

Calcium acts as a second messenger and transmits information from the extracellular side of the plasma membrane to targets within the cell [33, 45, 46]. In the case of Ca^{2+} signaling in astrocytes, however, the information usually arrives as a nonoscillatory stimulus at the plasma membrane and is translated into intracellular Ca^{2+} oscillations. For instance, glutamate concentration at the extracellular side of the astrocyte membrane determines the degree of activation of mGluRs and therefore can be directly linked to intracellular IP_{3} concentration [47]. It follows that in the L–R model, the level of IP_{3} can be thought as being directly controlled by glutamate signals impinging on the cell from its external environment. In turn, the level of IP_{3} determines the dynamics of intracellular Ca^{2+}. In physiological conditions glutamate-induced astrocyte Ca^{2+} signaling is synaptically evoked [2–4]. One can therefore think of the Ca^{2+} signal as being an encoding of information about the level of synaptically released glutamate and ultimately of synaptic activity. Notably, this information encoding can use AM, FM, or both modulations (AFM) of Ca^{2+} oscillations and pulsations.

We have recently shown that these encoding modes may actually depend on inherent cellular properties [18, 19]. In particular, the stronger the SERCA uptake with respect to Ca^{2+} efflux from the ER, the more pulsating and FM-like the encoding. A fast uptake by SERCAs, in fact, firmly counteracts CICR, so that higher Ca^{2+} levels are required for the onset of this latter one. When this happens, however the effects of CICR are large and the increase of intracellular Ca^{2+} is fast and remarkable. Accordingly, inactivation of IP_{3}R channels is also faster and basal Ca^{2+} levels are recovered rapidly. In these conditions, the IP_{3} level modulates the onset of CICR (through *m*_{∞}) thus setting the frequency of pulsation (FM encoding). On the contrary, the AM case is observed with weaker Ca^{2+} uptake by SERCAs. Weaker SERCA rates in fact allow for smoother oscillations whose amplitude is mainly dependent on the interplay between CICR onset and Ca^{2+}-dependent inactivation. Hence, the amplitude of oscillations in these latter conditions depends on IP_{3}, whereas their frequency does not, as it is essentially fixed by IP_{3}R channel recovery from Ca^{2+}-dependent inactivation [19].

From a dynamical systems perspective, AM and FM encoding are associated with well-distinct bifurcation diagrams. Amplitude modulations of Ca^{2+} oscillations are typically found when the system exhibits Hopf bifurcations only. In particular, when oscillations are born through a supercritical Hopf bifurcation at low IP_{3} concentration, then AM encoding exists (Fig. (Fig.2a–c).2a–c). Alternatively, if the Hopf bifurcation is subcritical, AFM might be found [18]. On the contrary, in FM (Fig. (Fig.2d–f),2d–f), the presence of a saddle-node homoclinic bifurcation accounts for pulsatile oscillations which arise at arbitrarily small frequency but with amplitude essentially independent of the IP_{3} value [19].

^{2+}oscillations can be generated by the Li–Rinzel model for CICR, depending on the value of

*K*

_{ER}, the Ca

^{2+}affinity of sarco-endoplasmic reticulum Ca

^{2+}-ATPase

**...**

Finally, it is important to note that the L–R model assumes that IP_{3} does not vary with time nor depends on the other variables (that is, its concentration *I*, in (5) and (6), is a parameter of the model). Yet examination of the underlying signaling pathways (Fig. (Fig.1)1) immediately hints that IP_{3} concentration indeed depends on both intracellular Ca^{2+} and extracellular glutamate, so that IP_{3} should be an additional variable in the model. Our aim in the present article is to devise a model that incorporates these dependencies.

### IP_{3} regulation: the *ChI* model

#### PLCδ production

In astrocytes, IP_{3} together with diacylglycerol (DAG) is produced by hydrolysis of phosphatidylinositol 4,5-bisphosphate (PIP_{2}) by two phosphoinositide-specific phospholipase C (PLC) isoenzymes, PLCβ and PLCδ [48]. The activation properties of these two isoenzymes are different and so, it is likely, are their roles. PLCβ is primarily controlled by cell surface receptors; hence, its activity is linked to the level of external stimulation (i.e., the extracellular glutamate) and as such, it pertains to the glutamate-dependent IP_{3} metabolism and will be addressed in the next section.

On the contrary, PLCδ is essentially activated by increased intracellular Ca^{2+} levels (Fig. (Fig.1d)1d) [49]. Structural and mutational studies of complexes of PLCδ with Ca^{2+} and IP_{3} revealed complex interactions of Ca^{2+} with several negatively charged residues within its catalytic domain [50–52], a hint of cooperative binding of Ca^{2+} to this enzyme. In agreement with these experimental findings, the PLCδ activation rate can be written as [27, 53]:

where the maximal rate of activation depends on the level of intracellular IP_{3}. Experimental observations show that high (>1 μM) IP_{3} concentrations inhibit PLCδ activity by competing with PIP_{2} binding to the enzyme [54]. Accordingly, assuming competitive binding [55], the maximal PLCδ-dependent IP_{3} production rate can be modeled as follows:

where is the inhibition constant of PLCδ activity.

Figure Figure33 shows the behavior of this term when Ca^{2+} and corresponding IP_{3} levels obtained from the bifurcation diagrams of the L–R model in Fig. Fig.22 are substituted into (7). We have set to a value that is close to the Ca^{2+} concentration of the lower bifurcation point. This allows us to translate the large-amplitude Ca^{2+} oscillations into oscillations of that could preserve the main AM/FM properties.

#### IP_{3} degradation

Two major IP_{3} degradation pathways have been described so far (Fig. (Fig.1b).1b). The first one is through dephosphorylation of IP_{3} by inositol polyphosphate 5-phosphatase (IP-5P). The other one occurs through phosphorylation of IP_{3} by the IP_{3} 3-kinase (IP_{3}-3K) and is Ca^{2+} dependent [63].

The rate of both IP-5P dephosphorylation (*v*_{5P}) and IP_{3}-3K phosphorylation (*v*_{3K}) of IP_{3} can be considered as of Michaelis–Menten type [56, 64, 65]. Therefore:

Since M [57, 66] and physiological levels of IP_{3} are in general below this value, IP-5P is likely not to be saturated by IP_{3}. It follows that the rate of IP_{3} degradation by IP-5P can be linearly approximated:

where is the linear rate of IP_{3} degradation by IP-5P and can be defined by parameters in (9) as .

In basal conditions, phosphorylation of IP_{3} by IP_{3}-3K is very slow. The activity of IP_{3}-3K is substantially stimulated by Ca^{2+}/calmodulin (CaM) via CaMKII-catalyzed phosphorylation (Fig. (Fig.1b)1b) [67]. However, other experimental reports have suggested that Ca^{2+}-dependent PKC phosphorylation of IP_{3}-3K could have inhibitory effects [68]. Notwithstanding, evidences for this latter possibility are contradictory [69]. Hence, for the sake of simplicity, we have chosen in the present model to consider the simplified case where only CaMKII-catalyzed phosphorylation of IP_{3}-3K is present (Fig. (Fig.11f).

Phosphorylation of IP_{3}-3K by active CaMKII (CaMKII*) only occurs at a single threonine residue [67, 70], therefore we can assume that [CaMKII*]. Activation of CaMKII is Ca^{2+}/CaM dependent and occurs in a complex fashion because of the unique structure of this kinase which is composed of ~12 subunits with three to four phosphorylation sites each [71]. Briefly, Ca^{2+} elevation leads to the formation of a Ca^{2+}–CaM complex (CaM^{+}) that may induce phosphorylation of some of the sites of each CaMKII subunit. CaMKII quickly and fully activates when two of these sites (at proximal subunits) are phosphorylated [72]. In spite of the occurrence of multiple CaM^{+} binding to the inactive kinase, experimental investigations showed that KII activation by CaM^{+} can be approximated by a Hill equation with unitary coefficient [58]. Hence, if we surmise the following kinetic reaction scheme for CaMKII phosphorylation:

it can be demonstrated that [CaMKII*] Hill(*C*^{4}, *K*_{D}) (see Appendix 2).

Accordingly, and the equation for IP_{3}-3K-dependent IP_{3} degradation reads:

Experimental observations show the existence of three regimes of IP_{3} metabolism [57]. At low [Ca^{2+}] and [IP_{3}] (<400 nM and <1 μM, respectively), IP-5P and IP_{3}-3K degrade roughly the same amounts of IP_{3}. Then, at high [Ca^{2+}] (≥400 nM) but low [IP_{3}] (≤8 μM), IP_{3} is predominantly metabolized by IP_{3}-3K. Eventually, for [IP_{3}] greater than 8 μM, when IP_{3}-3K activity saturates, IP-5P becomes the dominant metabolic enzyme, independently of [Ca^{2+}].

In our modeling, the third regime—corresponding to [IP_{3}] > 8 μM—exceeds the range of validity for the linear approximation of IP-5P degradation (10) and therefore cannot be taken into account. However, it can be shown that the first two regimes are sufficient to reproduce Ca^{2+} oscillations and pulsations, thus restricting the core features of IP_{3} metabolism to the maximal rates of IP_{3} degradation by IP_{3}-3K and IP-5P and to the Ca^{2+} dependence of IP_{3}-3K. In particular, by opportune choice of parameters such as , theoretical investigations showed that these two regimes are essentially generated by the Ca^{2+}-dependent Hill term in the expression of *v*_{3K} irrespectively of the assumption of Michaelis–Menten kinetics for IP_{3} dependence of IP_{3}-3K (Fig. (Fig.4).4). Accordingly, a linear approximation for *v*_{3K} such as:

where could also be considered instead of (12), in agreement with previous investigations found in the literature [26, 57].

**a**Experimental observations suggest the existence of three regimes of IP

_{3}metabolism: one for low [Ca

^{2+}] and [IP

_{3}] in which IP

_{3}-3K (Ca

^{2+}-dependent

*color*curves) and IP-5P (

*black*curve) activities are similar; an intermediate

**...**

Indeed, the behaviors of *v*_{3K} in (12) and (13) for IP_{3} and Ca^{2+} concentrations obtained from the corresponding Li–Rinzel bifurcation diagrams are qualitatively similar (Fig. (Fig.5).5). Moreover, the overall bifurcation diagrams are largely conserved (results not shown). The main quantitative difference is that the linear approximation yields stronger degradation rates. In particular, the IP_{3}-3K rate can be up to twofold higher in (13) than in (12). This is particularly marked when high [Ca^{2+}] is reached, such as in FM conditions (Fig. (Fig.5c–d).5c–d). Notwithstanding, the Michaelis–Menten constant of IP_{3}-3K for its substrate is experimentally reported to be M [56, 59] and it is likely that intracellular IP_{3} levels can reach such micromolar concentrations in vivo [73]. Therefore, in the following, we will keep the Michaelis–Menten formulation for *v*_{3K} (12).

Finally, experimental measurements show that for [Ca^{2+}] > 1 μM and low IP_{3} levels, the IP_{3}-3K activity exceeds that of IP-5P by almost 20-fold. In the model, this means that if *I* << *K*_{3} (i.e., , then *v*_{3K}≈20*v*_{5P}. Accordingly, we set the maximal degradation rates in the following such that .

#### Model analysis

In summary, our model of Ca^{2+} dynamics with endogenous IP_{3} metabolism is based on the two L–R equations ((5) and (6)), but the IP_{3} concentration (*I*) is now provided by a third coupled differential equation (summing the terms given by (7), (10), (12)):

Equation 14 together with (5) and (6) define our three-variable “*ChI*” model, whose name is composed of the letters denoting its state variables.

Consistency of the *ChI* model with respect to the L–R core model was sought by comparing two curves for pseudosteady states. First, we set and *C* → 0 in (14) and solved for *I* as a function of *C* in the resulting equation. In parallel, we set in (5) and solved for *I* as a function of *C* in the resulting equation as well. The two resulting *I*–*C* curves should be as similar as possible. Analysis showed that they are indeed relatively similar (Fig. 6) if one chooses , *K*_{D}≈H_{2}, *K*_{3}>H_{2}, where H_{1} and H_{2} denote Ca^{2+} and IP_{3} concentrations at the two Hopf bifurcations in the L–R bifurcation diagrams (Fig. (Fig.2).2). Such choice of parameters together with the others given in Table Table11 ensures the existence of Ca^{2+} and IP_{3} oscillations with amplitudes that are in agreement with those reported in the literature ([73]; see Fig. 3 of Online Supplementary Material).

_{3}metabolism with respect to the L–R core model can be tested as follows: at resting physiological conditions: , and

*C*→0 so that . Hence, for steady IP

_{3}values (

*I*

_{s}) such as , one gets

**...**

An important feature of our model is that despite the coupling between Ca^{2+} and IP_{3}, the equation for Ca^{2+} dynamics (5) does not contain parameters found within the equation for IP_{3} dynamics (14). This means that the equation of the *C*-nullcline does not change with respect to the L–R model. Because the shape of this nullcline is crucial for the encoding mode (see Fig. Fig.2a,2a, c), the occurrence of AM, FM, or AFM modes in the *ChI* model is essentially established by the parameters of the L–R core model.

The only possible way that IP_{3} metabolism could affect the encoding mode is by modulating the dynamics of the channel inactivation variable *h*. This mechanism is suggested by the projection of the surfaces for , , and (Fig. (Fig.7)7) onto the *C*–*I* plane for different values of *h* and *C* (Fig. (Fig.8).8). We note indeed that the *C*-nullcline depends on the value of *h* but not the *I*-nullcline. In contrast, both the *h*-nullcline and the *I*-nullcline change with *C*, which suggests that the coupling between Ca^{2+} and IP_{3} dynamics essentially occurs through *h*. We may expect that, since *h* sets the slow time scale of the oscillations, the effect of IP_{3} metabolism on Ca^{2+} dynamics in our model is mainly a modulation of the oscillation frequency. This aspect is further discussed in Sections 4 and 5, following the introduction in the next section of the last term of our model, namely the glutamate-dependent IP_{3} production.

## Modeling glutamate regulation of IP_{3} production: the G-*ChI* model

The contribution of glutamate signals to IP_{3} production can be taken into account as an additional production term in the IP_{3} equation of the above three-variable *ChI* model. The resulting new model is referred to as the “G-*ChI*” model.

Glutamate-triggered Ca^{2+} signals in astrocytes are mediated by group I and II mGluRs [74]. Metabotropic GluRs are G-protein coupled receptors associated with the phosphotidylinositol signaling-cascade pathway [75]. Although it is likely that the type of mGluRs expressed by astrocytes depends on the brain region and the stage of development [76], it seems reasonable to assume that such differences are negligible in terms of the associated second-messenger pathways [77, 78].

The G protein associated with astrocyte mGluRs is a heterotrimer constituted by three subunits: *α*, *β*, and *γ*. Glutamate binding to mGluR triggers receptor-catalyzed exchange of GTP from the G*βγ* subunits to the G*α* subunit. The GTP-loaded G*α* subunit then dissociates from the G protein in the membrane plane and binds to a colocalized PLCβ (Fig. (Fig.1a,1a, e). Upon binding to G*α*, the activity of PLCβ substantially increases, thus promoting PIP_{2} hydrolysis and IP_{3} production. Activation of PLCβ can therefore, to a first approximation, be directly linked to the number of bound mGluRs, and hence to the level of external stimulation. It follows that glutamate-dependent IP_{3} production can be written in the following generic form:

where is the maximal PLCβ rate that depends on the surface density of mGluRs and *R*(*γ*,*C* ) is the fraction of activated (bound) mGluRs. Experimental evidence shows that PLCβ activity (i.e., in (15)) is also dependent on intracellular Ca^{2+} [49]. Notwithstanding, such dependence seems to occur for [Ca^{2+}] > 10 μM, hence out of our physiological range [54]. Therefore, Ca^{2+} dependence of PLCβ maximal rate will not be considered here.

*R*(*γ*,*C* ) can be expressed in terms of extracellular glutamate concentration (*γ*) at the astrocytic plasma membrane, assuming a Hill-binding reaction scheme, with an exponent ranging between 0.5 and 1 [60]. In the current study, we choose 0.7, yielding:

In (16), *R*(*γ*,*C* ) is expressed as a Hill function with a midpoint that depends on glutamate and intracellular Ca^{2+} concentrations. This choice was motivated by the termination mechanism of PLCβ signaling that occurs essentially through two reaction pathways [48]: (a) reconstitution of the inactive G-protein heterotrimer due to the intrinsic GTPase activity of activated G*α* subunits and (b) PKC phosphorylation of the receptor, or of the G protein, or of PLCβ, or some combination thereof. We lump both effects into a single term, *K*_{γ}(*γ*,*C* ), such that the effective Hill midpoint of *R*(*γ*,*C* ) increases as PLCβ termination takes over, namely:

Here, *K*_{R} is the Hill midpoint of glutamate binding with its receptor whereas *K*_{p} measures the increment of the apparent affinity of the receptor due to PLCβ terminating signals.

Hill(*γ*^{0.7}, *K*_{R}) accounts for the intrinsic GTPase-dependent PLCβ activity termination, as this effect is linked to the fraction of activated G*α* subunits and therefore can be put in direct proportionality with the fraction of bound receptors. Hill instead accounts for PKC-related phosphorylation-dependent termination of PLCβ activity. Experimental data suggest that the target of PKC in this case is either the G protein or PLCβ itself [79]. Generally speaking, phosphorylation by PKC may modulate the efficiency of ligand-binding by the receptors, the coupling of occupied receptors to the G protein, or the coupling of the activated G protein to PLCβ [80]. All these effects indeed are lumped into (17), as explained below.

PKC is activated in a complex fashion (Fig. (Fig.1e).1e). Indeed, its activation by mere intracellular Ca^{2+} is minimal [81], while full activation is obtained by binding of the coactivator DAG. In agreement with this description, PKC activation can be approximated by a generic Hill reaction scheme, whereas Ca^{2+}-dependent PKC phosphorylation can be assumed Michaelis–Menten [79] so that . Remarkably, [DAG] can itself be related to intracellular Ca^{2+} concentration [82] so that [PKC*] can be rewritten as . Finally, *K*_{DAG}<<*K*_{π} [61, 81, 82] so that we can eventually approximate the product of the two Hill functions by that with the highest midpoint (see Appendix 1 for the derivation of this approximation). That yields: , which accounts for the second Hill function in (17).

To complete the model, it can be shown by numerical analysis of (17) that the term related to the GTPase-dependent PLCβ termination pathway, i.e., Hill(*γ*^{0.7}, *K*_{R}), can be neglected to a first approximation (Fig. (Fig.9).9). Hence can be simplified as :

Using (15), (16), and (18), our final expression for the glutamate-dependent IP_{3} production reads:

Substituting (19) into (14), we obtain

This equation, combined with (5) and (6), defines our G-*ChI* model of glutamate-dependent intracellular Ca^{2+} dynamics in astrocytes.

## Dynamical behaviors and coding modes of the G-*ChI* model

The dynamical features of the G-*ChI* model for different extracellular concentrations of glutamate can be appreciated by inspection of the bifurcation diagrams in Figs. Figs.1010 and and11.11. We note that the choice of , the maximal rate of glutamate-dependent IP_{3} production which is linked to the density of receptors on the extracellular side of the astrocyte membrane, can substantially influence the bifurcation structure of the model and the extent of the oscillatory range. Indeed, as decreases, the oscillatory range expands toward infinite glutamate concentrations, but the amplitude of oscillations concomitantly decreases (at least with regard to the IP_{3} concentration).

*ChI*model for FM-encoding sets of parameters.

**d**–

**i**In analogy with Fig. Fig.10,10, reduced values of , the maximal rate of PLCβ-dependent IP

_{3}production, extend to infinity the range of oscillations,

**...**

The extension of the oscillatory range is due to the shift toward infinity of the subcritical Hopf bifurcation at high glutamate concentrations (compare Fig. Fig.11a,11a, d). Notably, for some values of receptor density, there seems to be coexistence of oscillations and asymptotic stability at high concentrations of extracellular glutamate, depending on the state of the cell prior to the onset of stimulation (Fig. (Fig.1111d–f).

As decreases, degradation becomes progressively preponderant so that IP_{3} peak levels are lower and the IP_{3}R channels’ open probability is also reduced. Consequently, CICR is weaker and the increase of cytosolic Ca^{2+} is smaller. Then Ca^{2+}-dependent PKC activation is reduced and termination of PLCβ signaling by PKC-dependent phosphorylation is limited. Moreover, if saturation of receptors occurs (i.e., *R*(*γ*, *C* )≈1) and oscillations are observed in this case, it follows that higher extracellular glutamate concentrations cannot further affect the intracellular Ca^{2+} dynamics.

The value of at which intracellular Ca^{2+} dynamics locks onto stable oscillations also depends on , the strength of the endogen PLCδ-mediated IP_{3} production. To some extent, increasing decreases the minimal value above which oscillations appear, provided that CICR is strong enough to activate enough PLCδ to keep IP_{3} levels above the lower Hopf bifurcation (results not shown).

Coupling between IP_{3} and Ca^{2+} dynamics in the G-*ChI* model might have important implications for the encoding of the stimulus. Bifurcation diagrams in Figs. Figs.1010 and and1111 were derived using different sets of parameters that pertain respectively to AM and FM encoding in the *ChI* model as well as in the L–R core model (see Table Table11 and Fig. 3 in Online Supplementary Material). Notwithstanding, the applicability of these definitions to the G-*ChI* model might lead to some ambiguity.

We have previously assumed that AM (FM) encoding exists only if the amplitude (frequency) of oscillations (pulsations) throughout the oscillatory range can at least double with respect to its minimum value [19]. Here, if we consider the AM-derived bifurcation diagrams for Ca^{2+} and IP_{3} dynamics (Fig. (Fig.10),10), we note that AM is still found since oscillations occur with arbitrarily small amplitude for the supercritical Hopf point at lower stimulus intensity (Fig. (Fig.10a,10a, b, d, e). But the period of oscillations (Fig. (Fig.10c,10c, f) at the upper extreme of the oscillatory range is almost half that observed at the onset of oscillations at the lower Hopf point. Thus, FM also occurs. Notably, in such conditions, oscillations resemble pulsating dynamics. In other words, rather than pure AM encoding, as we could expect by a set of parameters that provides AM in the *ChI* model (Fig. 1a–c in Online Supplementary Material), it seems that, in the G-*ChI* model, Ca^{2+} oscillations become AFM encoding. Notably, IP_{3} dynamics appears to be always AFM encoding both in the AM (Fig. (Fig.10b,10b, e) and in the FM-derived bifurcation diagrams (Fig. (Fig.11b,11b, f, i).

Conversely, mere FM encoding is essentially preserved for Ca^{2+} dynamics derived from FM encoding sets of parameters in the *ChI* model, although a significant increase of the range of amplitudes of pulsations can be pointed out (compare Fig. Fig.11d11d with Fig. 3d in Online Supplementary Material). These observations indicate that the G-*ChI* model accounts either for FM or AFM encoding Ca^{2+} oscillations, which are, however always coupled with AFM encoding IP_{3} oscillations. In addition, they provide further support to the above-stated notion that IP_{3} metabolism could consistently modulate the frequency of Ca^{2+} pulsating dynamics more than their amplitude (see Section 2.3.3).

On the contrary, the amplitude and shape of IP_{3} oscillations appear to be dramatically correlated with those of Ca^{2+} oscillations, as a consequence of the numerous Ca^{2+}-dependent feedbacks on IP_{3} metabolism. Smooth Ca^{2+} oscillations such as those obtained in AM-like conditions (Fig. (Fig.12a,12a, AM) are coupled with small zigzag IP_{3} oscillations (Fig. (Fig.12b,12b, AM). Under FM conditions instead, pulsating large-amplitude Ca^{2+} variations (Fig. (Fig.12a,12a, FM) can be lagged by analogous IP_{3} oscillations (Fig. (Fig.12b,12b, FM), with the difference that whereas Ca^{2+} pulsations are almost fixed in their amplitude, IP_{3} ones can substantially vary.

**a**,

**b**Examples of forced burst oscillations exhibited by the G-

*ChI*model, under

**c**a square-wave stimulus protocol. This figure illustrates how stationary glutamate stimulations are encoded as oscillations and pulsations of the second messengers Ca

^{2+}

**...**

Simulations of physiologically equivalent glutamate stimulation and associated astrocyte Ca^{2+}–IP_{3} patterns are shown in Fig. Fig.13.13. Real multi-array electrode-recording data were considered as inputs of a single glutamatergic synapse (modeled as in Tsodyks and Markram [83]) and a fraction of the released glutamate was assumed to impinge on the astrocyte described by our model. We may notice that, from the stimulus up to Ca^{2+} dynamics, the smoothness of the patterns seems to increase. Indeed, the highly jagged glutamate stimulus turns into a less indented IP_{3} signal which is coupled with even smoother Ca^{2+} oscillations. Depending on the inherent cellular properties (Fig. (Fig.13,13, for example, considers two cases associated with different SERCA Ca^{2+} affinities), the difference of smoothness between IP_{3} and Ca^{2+} can be dramatic, more likely in the case of FM encoding Ca^{2+} pulsations (compare Fig. Fig.13a–b,13a–b, AM and FM).

## Discussion

Calcium dynamics in astrocytes can be driven by extracellular signals (such as the neurotransmitter glutamate) through regulation of the intracellular IP_{3} levels. Therefore, a prerequisite for unraveling the response of astrocytes to such signals is a thorough understanding of the complex IP_{3}-related metabolic pathways that regulate intracellular Ca^{2+} dynamics. Here, we have devised and studied a model for agonist-dependent intracellular Ca^{2+} dynamics that captures the essential biochemical features of the complex regulatory pathways involved in glutamate-induced IP_{3} and Ca^{2+} oscillations and pulsations. Our model is simple, yet it retains the essential features of the underlying physiological processes that constitute the intricate IP_{3} metabolic network.

More specifically, the equation for IP_{3} dynamics is a central component of our model because of the large number of metabolic reactions that it accounts for and because coupling with intracellular Ca^{2+} dynamics is resolved through complex feedback mechanisms. Production of IP_{3} depends on the agonist/receptor-dependent PLCβ activation as well as on the endogenous agonist-independent contribution of PLCδ because both isoenzymes are found in astrocytes [48].

We linked the relative expression of these two isoenzymes to the expression of PKC and to the strength of PLCβ regulation by PKC. Indeed, Ca^{2+}-dependent PKC activation can phosphorylate the receptor or PLCβ or a combination thereof, leading to termination of IP_{3} production [79]. In astrocytes, this mechanism has been suggested to limit the duration of Ca^{2+} oscillations, thus defining their frequencies [82]. In agreement with this idea, a stronger PKC-dependent inhibition of PLCβ shrank the oscillatory range in our model astrocyte and led to the progressive loss of long-period oscillations.

In our model, the PKC-dependent inhibition of PLCβ is counteracted if PLCδ expression is high enough to support high IP_{3} production levels and the resulting release of Ca^{2+} from the intracellular stores. This observation raises the possibility of phase-locked Ca^{2+} oscillations under conditions of intense stimulation. Phase-locked Ca^{2+} oscillations were also found in other models of agonist-dependent intracellular Ca^{2+} dynamics [84–86] and are often associated with pathological conditions [87, 88]. In our model, persistent pulsating Ca^{2+} dynamics that are essentially independent of the level of stimulation are observed for weak maximal rates of IP_{3} production by PLCβ (Figs. (Figs.10d–e,10d–e, 11g–h). In astrocytes, such persistent oscillations could also be interpreted as a fingerprint of pathological conditions [1, 89]. In fact, a decay of PLCβ activity is likely to occur, for instance, if the density of effective metabotropic receptors in the astrocytic plasma membrane decreases, such as in the case of epileptic patients with Ammon’s horn sclerosis [90].

We note that, although focusing on stimulus-triggered Ca^{2+} oscillations, our study also hints, at a possible link between modulation of frequency and amplitude of Ca^{2+} pulsations and spontaneous Ca^{2+} dynamics. Recently, it has been shown that the interpulse interval of the spontaneous Ca^{2+} oscillations is inherently stochastic [91, 92]. In particular, experimental observations are compatible with model studies of a local stochastic nucleation mechanism that is amplified by the spatial coupling among IP_{3}R clusters through Ca^{2+} or IP_{3} diffusion [92, 93]. Our analysis may provide meaningful clues to identify what factors and processes within the cell could affect the rate of wave nucleation. More specifically, we may predict that putative intracellular IP_{3} dynamics could affect the statistics of Ca^{2+} interpulse intervals not only in terms of spatial coupling among IP_{3}R clusters by means of intracellular IP_{3} gradients but also by modulation of either the recovery from Ca^{2+} inhibition or the progressive sensitization of IP_{3}Rs by Ca^{2+} [94]. The resulting scenario therefore would still be that of a local stochastic nucleation mechanism amplified by IP_{3}R spatial coupling, but the local IP_{3}R and SERCA parameters would vary according to the biochemical regulation system presented in the current work.

A critical question in experiments is the identification of the mechanism that drives IP_{3} oscillations and pulsations [57, 95–97]. In our model, self-sustained IP_{3} oscillations are brought about by the coupling of IP_{3} metabolism with Ca^{2+} dynamics. In other words, our model can be considered as a self-consistent astrocytic generator of Ca^{2+} dynamics. This might have broad implications for astrocyte encoding of information and neuron–glia communication.

We previously demonstrated that modulation by astrocytes of synaptic information transfer could account for some of the peculiar dynamics observed in spontaneous activity of cultured cortical networks [13]. In particular, a simple neuron–glia circuit composed of an autaptic neuron “talking” with a proximal astrocyte could serve as a self-consistent oscillator when fed by weak external signals. The results presented in the current study suggest an alternative, more robust, way (independent of synaptic architecture) to form glia-based self-consistent oscillators. The relative contribution and significance of either the astrocytic or the IP_{3}-based hypotheses to the spontaneous network activity need to be assessed by future combined experiments and modeling. Meanwhile, the analysis of our present model suggests that, in astrocytes, different second messenger molecules are engaged in an intricate dialogue, likely meaning that those non-neural cells might be crucially important for deciphering some of the enigmas of neural information processing.

Another significant prediction of our model is that IP_{3} dynamics is essentially AFM, and Ca^{2+} oscillations/pulsations are inherently FM encoding, that is, they can be either FM or AFM but not AM [18, 19, 98]. In FM, Ca^{2+} oscillations resemble pulses. In contrast, in AFM, their shape is smoother and necessarily depends on the stimulus dynamics.

The assumption that IP_{3} oscillations are always AFM encoding could provide an optimal interface between agonist stimuli and intracellular Ca^{2+} signals. The stimuli impinge on the cell in the form of trains of pulses or bursts of pulses and information is carried in the timing of these pulses rather than in their amplitude [99]. AFM features in IP_{3} signals could perfectly match these stimuli, embedding the essential features of the spectrum of the signal into the spectrum of the IP_{3} transduction. Hence, IP_{3} signaling with FM features could offer an efficient way to keep the essence of the information of the stimulus. On the other hand, because Ca^{2+} signals are triggered primarily by sufficiently ample elevations of IP_{3} [35], the coexistence of AM features within the IP_{3} signal seems to be a necessary prerequisite in order to trigger CICR.

The fact that coupling of IP_{3} metabolism with CICR does not allow pure AM encoding is in general agreement with experimental data on intracellular Ca^{2+} signaling in several cells [100, 101] including astrocytes [3]. Notwithstanding, the possibility of AFM-encoding Ca^{2+} oscillations has recently come up as a reliable alternative mechanism to explain gliotransmitter exocytosis, which is dependent on a specific agonist that triggers astrocyte Ca^{2+} dynamics [21, 102].

The above could be relevant to understanding the origin of the integrative properties of Ca^{2+} signaling in astrocytes [103]. Our analysis in fact shows that such properties could result from at least two steps of integration, one at the transduction of the agonist signal into IP_{3} signal and the other at the cross-coupling between the IP_{3} and Ca^{2+} signals. Indeed, AFM-encoding IP_{3} dynamics could deploy smoothing of the highly indented agonist stimulus, thus hinting at possible integrative properties for IP_{3} signals (Fig. (Fig.13).13). On the other hand, the associated Ca^{2+} patterns look even smoother, suggesting a further integration step that likely relies only on the inherent features of CICR.

Below is the image is a link to a high resolution version

^{(23K, gif)}

The product of two Hill functions (*a-b*) with sufficiently distant midpoints is equivalent to the Hill function with the largest midpoint (*c*). Namely: Hill(*x*, *K*_{1}) · Hill(*x*, *K*_{2}) ≈ Hill(*x*, *K*_{2}) where *K*_{1}<<*K*_{2}. Midpoints are marked by vertical dashed lines; *K*_{1}: *red*; *K*_{2}: *blue*. (GIF 22.5KB)

^{(147K, eps)}

^{(51K, gif)}

Hill functions of Hill functions (*a-b*) can also be approximated by Hill functions. (*c-d*) Hill (Hill(*x*, *K*_{2}), *K*_{1}) = (1+*K*_{1})^{−1}. Hill(*x*, *K*_{1}*K*_{2}(1+*K*_{1})^{−1}). In this case the midpoint of the resulting Hill function depends on the specific values of the midpoints of the original Hill functions considered in the composition of the Hill-of-Hill function. (*e-h*) Hill(*x*, *K*_{1} · Hill (*x*, *K*_{2})) = (*x* + *K*_{2})/(*x* + *K*_{1} + *K*_{2}) = Hill (*x*, (*K*_{1} + *K*_{2})) + *f*(*x*), where *f*(*x*) = *K*_{2}/(*x* + *K*_{1} + *K*_{2}). Notably, *f*(*x*0) = *K*_{2}/(*K*_{1} + *K*_{2}) whereas *f*(*x*∞) ≈ 0, so that the resulting Hill curve is essentially comprised within the interval [*K*_{2}/(*K*_{1} + *K*_{2}),1). (GIF 50.6KB)

^{(285K, eps)}

^{(45K, gif)}

Bifurcation diagrams for a modified *ChI* model and prototypical sets of (*a-c*) AM-encoding and (*d-f*) FM-encoding L-R parameters. The bifurcation diagrams were computed after introduction into the *ChI* model of the rate of glutamate-dependent IP_{3} production, *v*_{glu}, as a free bifurcation parameter, namely . This figure shows that the *ChI* model can still display oscillations in presence of an external non-specific bias of IP_{3} production. This is a first suggestion that the corresponding glutamate-dependent G-*ChI* model may also display oscillations. The parameters are taken from Table 1. (GIF 44.6KB)

^{(191K, eps)}

## Acknowledgements

The authors wish to thank Vladimir Parpura, Giorgio Carmignoto, and Ilyia Bezprozvanny for insightful conversations. V. V. acknowledges the support of the U.S. National Science Foundation I2CAM International Materials Institute Award, Grant DMR-0645461. This research was supported by the Tauber Family Foundation, by the Maguy-Glass Chair in Physics of Complex Systems at Tel Aviv University, by the NSF-sponsored Center for Theoretical Biological Physics (CTBP), grants PHY-0216576 and 0225630, and by the University of California at San Diego.

#### Appendix 1

For the sake of simplicity, we have adopted throughout the text the following notation for the generic Hill function:

where *n* is the Hill coefficient and *K* is the midpoint of the Hill function, namely the value of *x* at which Hill.

It can be shown that the product of two Hill functions can be approximated by the Hill function with the greatest midpoint, when the two midpoints are distant enough from each others, that is:

if and only if *K*_{1} << *K*_{2} (Fig. 1, Online Supplementary Material). Indeed, under such conditions, Hill only when *x* >> *K*_{1}, hence

This result can be extended to the product of *N* Hill functions, that is:

provided that *K*_{1}<<*K*_{2} <<...<<*K*_{N}.

Notably, the product of Hill function is not the only case in which a functions composed by Hill functions can be approximated by a mere Hill function: other examples are given by functions of the type or (see Fig. 2 in Online Supplementary Material).

#### Appendix 2

We seek an expression for [CaMKII*] based on the following kinetic reaction scheme:

Let us first consider the reaction chain (22). We can assume that the second step is very rapid with respect to the first one [58, 104] so that generation of CaMKII* is in equilibrium with CaMKII consumption, namely:

Then, under the hypothesis of quasisteady state for CaMKII, we can write:

It follows that incorporation of (23) into (24) leads to:

where . Defining as the total kinase II concentration and assuming it constant, we can rewrite (25) as follows:

with .

## Footnotes

An erratum to this article can be found at http://dx.doi.org/10.1007/s10867-009-9182-8

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