G protein-coupled receptor-mediated calcium signaling in astrocytes
aa r X i v : . [ q - b i o . S C ] J a n G protein-coupled receptor-mediated calcium signaling inastrocytes
Maurizio De Pitt`aEPI BEAGLE, INRIA Rhˆone-Alpes, Villeurbanne, FranceEshel Ben-Jacob † School of Physics and Astronomy, Tel Aviv University, Ramat Aviv, IsraelHugues BerryEPI BEAGLE, INRIA Rhˆone-Alpes, Villeurbanne, FranceJanuary 23, 2018
Abstract
Astrocytes express a large variety of G protein-coupled receptors (GPCRs) which mediatethe transduction of extracellular signals into intracellular calcium responses. This transduc-tion is provided by a complex network of biochemical reactions which mobilizes a wealthof possible calcium-mobilizing second messenger molecules. Inositol 1,4,5-trisphosphate isprobably the best known of these molecules whose enzymes for its production and degra-dation are nonetheless calcium-dependent. We present a biophysical modeling approachbased on the assumption of Michaelis-Menten enzyme kinetics, to effectively describe GPCR-mediated astrocytic calcium signals. Our model is then used to study different mechanismsat play in stimulus encoding by shape and frequency of calcium oscillations in astrocytes.
Calcium signaling is the most common measured readout of astrocyte activity in response tostimulation, be it by synaptic activity, by neuromodulators diffusing in the extracellular milieu,or by exogenous chemical, mechanical or optical stimuli. In this perspective, the individualastrocytic Ca transient is thought, to some extent, as an integration of the triggering stimulus(Perea and Araque, 005a), and is thus regarded as an encoding or decoding of this stimulus,depending on the point of view (Carmignoto, 2000; De Pitt`a et al., 2013).Multiple and varied are the spatiotemporal patterns of Ca elevations recorded from as-trocytes in response to stimulation, each possibly carrying its own encoding (Bindocci et al.,2017). Insofar as different encoding modes could correspond to different downstream signal-ing, including gliotransmission and thereby regulation of synaptic function, understanding thebiophysical mechanisms underlying rich Ca dynamics in astrocytes is crucial.Calcium-induced Ca release (CICR) from the endoplasmic reticulum (ER) is arguablythe best characterized mechanism of Ca signaling in astrocytes (Zorec et al., 2012). It en-sues from nonlinear properties of Ca channels which are found on the ER membrane andare gated by the combined action of cytosolic Ca and the second messenger molecule inositol1,4,5-trisphosphate (IP ) (Shinohara et al., 2011, see also Chapters 2–4). This second messen-ger molecule can be produced by the astrocyte either spontaneously or, notably, in response † Deceased June 5, 2015.
1o activation by extracellular insults activation of G protein-coupled receptors (GPCRs) foundon the cell’s plasma membrane (Parri and Crunelli, 2003; Panatier et al., 2011; Volterra et al.,2014). Hence, IP together with these receptors, can be regarded as integral componentsof the interface whereby an astrocyte transduces extracellular insults into Ca responses(Marinissen and Gutkind, 2001). Characterizing this interface is thus an essential step in ourunderstanding of the emerging complexity of Ca signals, and we devote this chapter to thispurpose. In the first part of the chapter, we will present a concise framework to model in-tracellular IP signaling in astrocytes. This framework is general and can easily be extendedto include additional biological details, such as for example, the regulation of GPCR bindingefficiency by protein kinase C. Some of the models presented in this chapter are also subjectedto revision and comparison with other astrocyte models in Chapters 16 and 18. dynamics production G protein-coupled receptors form a large family of receptors which owe their name to theirextensively studied interaction with heterotrimeric G proteins (composed of an α , β and γ sub-unit) which undergo conformational changes that lead to the exchange of GDP for GTP, boundto the α -subunit, following receptor activation. Consequently, the G α - and G βγ -subunits stim-ulate enzymes thereby activating or inhibiting the production of a variety of second messengers(Marinissen and Gutkind, 2001).Among all GPCRs, those that contain the G α q subunit are linked with the cascade of chem-ical reactions that leads to IP synthesis. There, the G α q subunit promotes activation of theenzyme pospholipase C β (PLC β ) which hydrolizes the plasma membrane lipid phosphatidyli-nositol 4,5-bisphosphate (PIP ) into diacylglycerol (DAG) and IP (Rebecchi and Pentyala,2000). Examples of such receptors expressed by astrocytes ex vivo and in vivo are the type Imetabotropic glutamate receptor 1 and 5 (mGluR1/5) (Wang et al., 2006; Sun et al., 2013), thepurinergic receptor P Y (Jourdain et al., 2007; Di Castro et al., 2011; Sun et al., 2013), themuscarinic receptor mAchR1 α (Takata et al., 2011; Chen et al., 2012; Navarrete et al., 2012)and the adrenergic α receptor (Bekar et al., 2008; Ding et al., 2013). While these receptorsbind different agonists, and likely display receptor-specific binding kinetics, they all share thesame downstream signaling pathway and therefore may be modeled in a similar fashion.Several are the available models for G α q -containing receptors, and the choice of what modelto use rather than another depends on the level of biological detail and the questions one isinterested in. Here our focus is on the rate of IP production upon activation of these receptors,so we wish to keep as simple as possible the description of the reactions that regulate theactivation of PLC β by α q , β and γ subunits. This is possible, assuming that these reactionsare much faster than the downstream ones that result in IP production. In this case, a quasisteady-state approximation (QSSA) holds whereby, in the series of reactions that leads fromreceptor agonist binding to activation of PLC β , the intermediate reactions involving the threereceptor’s subunits are at equilibrium on the time scale of the production of activated PLC β .Accordingly, assuming that on average the receptor at rest (R) requires n molecules of an agonist(A) to promote activation of PLC β (R * ) at rate O N , we can writeR + n A O N R ∗ (1)We further make another assumption: that the cascade of reactions that leads to GPCR-mediated IP synthesis has a Michaelis-Menten kinetics (see Appendix A.2), so the IP pro-duction by PLC β ( J β ) can be taken proportional to the fraction of bound receptors, defined2s Γ A = [R ∗ ] / [R] T , with [R] T –– [R] + [R * ] being the total receptor concentration at the site ofIP production, i.e., J β = O β · Γ A (2)In the above equation O β is the maximal rate of IP production by PLC β and lumps informa-tion on receptor surface density as well as on the size of the PIP reservoir. Importantly, thesetwo quantities may not be fixed, insofar as receptors are subjected to desensitization, inter-nalization and recycling, and the reservoir of PIP could also be modulated by cytosolic Ca and IP (Rhee and Bae, 1997). The reader interested in modeling these aspects may refer toLemon et al. (2003). In the following, we will assume O β constant for simplicity.To seek an expression for J β , termination of PLC β signaling has to be considered. With thisregard, as illustrated in Figure 1A, there are two possible pathways whereby IP production byPLC β ends (Rebecchi and Pentyala, 2000). One is by reconstitution of the inactive G proteinheterotrimer, and coincides with unbinding of the agonist from the receptor, due to the intrinsicGTPase activity of the activated G α q subunit. The other is by phosphorylation of the receptor,the G α q subunit, PLC β or some combination thereof by conventional protein kinases C (cPKC)(Ryu et al., 1990; Codazzi et al., 2001). This phosphorylation modulates either receptor affinityfor agonist binding, or coupling of the bound receptor with the G protein, or coupling of theactivated G protein with PLC β , ultimately resulting in receptor desensitization (Fisher, 1995).Denoting by cPKC * the active, receptor-phosphorylating kinase C, termination of PLC β -mediated IP production can then be modeled by the following pair of chemical reactions:R ∗ Ω N R + n A (3)cPKC ∗ + R ∗ O KR Ω KR cPKC ∗ − R ∗ Ω K cPKC ∗ + R + n A (4)From equations 3–4 we have:dR ∗ d t = O N [A] n [R] − Ω N [R ∗ ] − O KR [cPKC ∗ ][R ∗ ] + Ω KR [cPKC ∗ − R ∗ ] (5)d[cPKC ∗ − R ∗ ]d t = O KR [cPKC ∗ ][R ∗ ] − (Ω KR + Ω K )[cPKC ∗ − R ∗ ] (6)Assuming that production of the intermediate kinase-receptor complex is at quasi steady statein reaction 4, i.e. d[cPKC ∗ − R ∗ ] / d t ≈
0, provides (equation ?? )[cPKC ∗ − R ∗ ] = O KR Ω KR + Ω K [cPKC ∗ ][R ∗ ] (7)Then, substituting this latter equation in equation 5 givesdR ∗ d t = O N [A] n [R] − Ω N [R ∗ ] − O KR (cid:18) − Ω KR Ω KR + Ω K (cid:19) [cPKC ∗ ][R ∗ ]= O N [A] n [R] − Ω N [R ∗ ] − O K [cPKC ∗ ][R ∗ ] (8)where we defined O K = O KR (1 − Ω KR / (Ω KR + Ω K )).To retrieve an equation for [cPKC * ], we consider the fact that activation of cPKC requiresbinding to the kinase of free cytosolic Ca ( C ) and DAG, but only if Ca binds first, cPKC canget sensibly activated by DAG (Oancea and Meyer, 1998). Accordingly, the following sequentialbinding reaction scheme for cPKC activation may be assumed:cPKC + Ca O KC Ω KC cPKC ′ (9)3PKC ′ + DAG O KD Ω KD cPKC ∗ (10)where cPKC is the inactive kinase, and cPKC ′ denotes the Ca -bound kinase complex.By QSSA in reaction 4 it follows that the available activated kinase approximately equalsto [cPKC * ] T = [cPKC * ] + [cPKC * – R * ] ≈ [cPKC * ]. Moreover, it can be assumed that onlya small fraction of cPKC ′ is bound by DAG so that [cPKC ∗ ] ≪ [cPKC ′ ]. In this fashion,the available cPKC, denoted by [cPKC] T , can be approximated by [PKC] T ≈ [PKC] + [PKC ′ ].Accordingly, solving reactions 9 and 10 for [PKC ∗ ] provides[cPKC ∗ ] = (cid:0) [cPKC ∗ ] + [cPKC ′ ] (cid:1) · H ([DAG] , K KD ) ≈ [cPKC ′ ] · H ([DAG] , K KD )= [cPKC] T · H ( C, K KC ) · H ([DAG] , K KD ) (11)where K KD = Ω KD /O KD and K KC = Ω KC /O KC , and H ( x, K ) denotes the Hill function x/ ( x + K ) (Appendix A.1). In practice the activation of the kinase consists of two sequentialtranslocations to the plasma membrane of its C2 and C1 domains (Oancea and Meyer, 1998).The translocation of C2 is regulated by Ca whereas that of C1 is by DAG. In this processhowever, experiments showed that the initial translocation of C2 is the rate limiting step forkinase activation (Shinomura et al., 1991), inasmuch as C1 translocation rapidly follows thatof C2 (Codazzi et al., 2001). This agrees with the notion that the cPKC affinity for DAGis regarded to be much higher than the affinity of the kinase for Ca , i.e. K KD ≪ K KC (Nishizuka, 1995). Since the product of two Hill functions with widely separated constantscan be approximated by the Hill function with the largest constant (De Pitt`a et al., 2009),equation 11 can be rewritten as[cPKC ∗ ] ≈ [cPKC] T · H ( C, K KC ) (12)which, once replaced in equation 8, gives:d[R ∗ ]d t = O N [A] n [R] − Ω N (cid:18) O K [cPKC] T Ω N H ( C, K KC ) (cid:19) [R ∗ ] (13)Finally, dividing both left and right terms in the above equation by [R] T , equation 13 can berewritten as dΓ A d t = O N [A] n (1 − Γ A ) − Ω N (1 + ζ · H ( C, K KC )) Γ A (14)where ζ = O KC [cPKC] T / Ω N quantifies the maximal receptor desensitization by cPKC. In theapproximation that receptor binding and activation is much faster than the effective PLC β -mediated IP production, Γ A can be solved for the steady state. In this fashion, IP productionby PLC β in equation 2 becomes J β = O β · H n (cid:16) [A] , ( K N (1 + ζ H ( C, K KC ))) n (cid:17) (15)where K N = Ω N /O N . The Hill coefficient n denotes cooperativity of the binding reaction ofthe agonist with the receptor and is both receptor and agonist specific. For example, glutamatebinding to subtype 1 mGluRs, such as those expressed by astrocytes (Gallo and Ghiani, 2000),is characterized by negative cooperativity and found in association with a Hill coefficient of n = 0 . − .
88 (Suzuki et al., 2004). On the contrary, binding of ATP to P Y Rs of dorsal spinalcord astrocytes from rats is characterized instead by almost no cooperativity and n = 0 . − .2 IP production by receptors with α subunits other than q-type A series of other astrocytic GPCRs, that traditionally associate with non- α q subunits, have alsobeen reported to mediate IP -triggered CICR, both in situ and in vivo. These include G α i / o -coupled GABA B receptors (Kang et al., 1998; Serrano et al., 2006; Mariotti et al., 2016), endo-cannabinoid CB receptors (Navarrete and Araque, 2008; Min and Nevian, 2012), adenosiner-gic A receptors (Crist´ov˜ao-Ferreira et al., 2013), adrenergic α receptors (Bekar et al., 2008),and dopaminergic D receptors (Jennings et al., 2017); as well as G α s -coupled receptorslike adenosine A receptors (Crist´ov˜ao-Ferreira et al., 2013), and dopamine D receptors(Jennings et al., 2017). α i / o and α s subunits are not expected to be linked with IP synthe-sis (Marinissen and Gutkind, 2001), rather they respectively inhibit or stimulate intracellularproduction of cAMP. Therefore the mechanism whereby these receptors could also promotemobilization of Ca from IP -sensitive ER stores remains a matter of investigation.One obvious possibility is that some of these receptors could be atypical in astrocytes and alsobe coupled with G α q , as it seems the case for example of astrocytic CB Rs in the hippocampus(Navarrete and Araque, 2008) and in the basal ganglia (Mart´ın et al., 2015). Biased agonismcould also be another possibility since the spatiotemporal pattern of agonist action on GPCRscould be quite different depending on agonist-binding kinetics of the receptor, especially ifagonists differentially engage dynamic signalling and regulatory processes (Overington et al.,2006), such as in the likely scenario of synapse-astrocyte interactions (Heller and Rusakov,2015). However, there is not yet direct structural evidence for distinct receptor conformationslinked to specific signals such as distinct G protein classes, and future studies are requiredto compare crystal structures of astrocytic GPCRs bound to biased and unbiased ligands toestablish these relationships (Violin et al., 2014).Alternatively, other signaling pathways mediated by cAMP that result in CICR could alsobe envisaged. In particular, Doengi et al. (2009) reported that GABA-evoked astrocytic Ca events in the olfactory bulb are fully prevented by blockers of astrocytic GABA transporters(GATs), but only partially by GABA B antagonists. GAT activation leads to an increase ofintracellular Na + , since this ion is cotransported with GABA, and such increase indirectlyinhibits the Na + /Ca exchanger on the plasma membrane. In turn, the ensuing Ca increasecould be sufficient to induce Ca release from internal stores by stimulation of endogenous IP production (Losi et al., 2014, see the following Section). This possibility is further corroboratedby the observation that astrocytic GATs could indeed be inhibited or stimulated respectivelyby A Rs or A Rs (Crist´ov˜ao-Ferreira et al., 2013).Yet other mechanisms could be at play for different receptors. Dopaminergic receptors forexample could either increase (D receptors) or decrease (D receptors) intracellular Ca levels in astrocytes (Jennings et al., 2017). This could indeed be explained assuming a possibleaction of these receptors on GATs which, similarly to adenosinergic receptors, could respectivelyincrease or reduce GABA/Na + cotransport into the cell, ultimately promoting or inhibitingCICR according to what was suggested for GABA B Rs. However there is also evidence thatnontoxic levels of dopamine could be metabolized by monoamine-oxidase in cultured astrocytes,resulting in the production of hydrogen peroxide (Vaarmann et al., 2010). This reactive oxygenspecies ultimately activates lipid peroxidation in the neighboring membranes which in turntriggers PLC-mediated IP production and CICR. Overall these different scenarios unraveladditional complexity in the possible mechanisms of GPCR-mediated CICR in astrocytes andcall for future modeling efforts that are beyond the scope of this chapter.5 .3 Endogenous IP production Phospholipase C δ (PLC δ ) is the enzyme responsible of endogenous IP production in as-trocytes, that is IP production that does not require external (i.e. exogenous) stimulation(Ochocka and Pawelczyk, 2003; Suh et al., 2008). The specific catalytic activity of this enzymein the presence of cytosolic Ca is 50- to 100-fold greater than Ca -stimulated activity ofPLC β in the absence of activating G protein subunits (Rebecchi and Pentyala, 2000), suggest-ing that PLC δ is prominently activated by increases of intracellular Ca (Rhee and Bae, 1997).Figure 1B exemplifies the biochemical network associated with PLC δ activation. Structuraland mutational studies of PLC δ complexes with Ca and IP , revealed complex interactionsof Ca with several negatively charged residues within the PLC δ catalytic domain (Essen et al.,1996, 1997; Rhee and Bae, 1997), hinting cooperative binding of at least two Ca ions with thisenzyme (Essen et al., 1997). In agreement with these experimental findings, we model PLC δ -mediated IP production ( J δ ) as (Pawelczyk and Matecki, 1997; H¨ofer et al., 2002): J δ = ˆ J δ ( I ) · H ( C, K δ ) (16)where H ( C, K δ ) denotes the Hill function of C with coefficient 2 and affinity K δ (Appendix B),and ˆ J δ ( I ) is the maximal rate of IP production by PLC δ which depends on intracellular IP ( I ). Experiments revealed that high IP concentrations, i.e. > µ m , inhibit PLC δ activityby competing with PIP binding to the enzyme (Allen and Barres, 2009). Accordingly, themaximal PLC δ dependent IP production rate can be modeled byˆ J δ ( I ) = O δ Iκ δ = O δ (1 − H ( I, κ δ )) (17)where O δ is the maximal rate of IP production by PLC δ and κ δ is the inhibition constantof PLC δ activity. degradation There are two pathways for IP degradation in astrocytes. The first one is by dephosphorylationof IP by inositol polyphosphate 5-phosphatase (IP-5P). The other one occurs through phos-phorylation of IP by the IP dependent butin opposite ways: while the activity of IP
3K is stimulated by cytosolic Ca (Communi et al.,1997), IP-5P is inhibited instead (Communi et al., 2001) (Figure 2A). Thus, depending onthe Ca concentration in the cytoplasm, different mechanisms of IP degradation could ex-ist (Sims and Allbritton, 1998). Moreover, IP-5P-mediated IP degradation could also be in-hibited by competitive binding of inositol 1,3,4,5-tetrakisphosphate (IP ) produced by IP -3K-mediated IP phosphorylation (Connolly et al., 1987; Erneux et al., 1998), thereby makingthe two degradation pathways interdependent (Hermosura et al., 2000). However, we will notconsider this aspect any further, since modeling of this reaction pathway requires a detailedconsideration of the complex metabolic network underpinning degradation of the large familyof inositol phosphates (Communi et al., 2001; Irvine and Schell, 2001). The reader interestedin these aspects may refer to Dupont and Erneux (1997) for a sample modeling approach to theproblem.Both IP-5P-mediated dephosphorylation ( J P ) and IP ( J K ) can be described by Michaelis-Menten kinetics (Irvine et al., 1986; Togashi et al., 1997),i.e., J P = ˆ J P · H ( I, K ) (18)6 K = ˆ J K ( C ) · H ( I, K ) (19)Since K P > µ m (Verjans et al., 1992; Sims and Allbritton, 1998), and such high IP con-centrations are unlikely to be physiological (Lemon et al., 2003; Kang and Othmer, 2009), theactivity of IP-5P can be assumed far from saturation. Accordingly, the IP degradation rateby IP-5P can be linearly approximated by (Stryer, 1999): J P ≈ Ω P · I (20)where Ω P = ˆ J P /K is the maximal rate of IP-5P-mediated IP degradation in the linearapproximation.IP phosphorylation by IP
3K is regulated in a complex fashion (Figure 2A). For resting con-ditions, when intracellular IP and Ca concentrations are below 0.1 µ M, (Parpura and Haydon,2000; Mishra and Bhalla, 2002; Kang and Othmer, 2009), it is very slow. On the other hand,as Ca increases, IP
3K activity is substantially stimulated by its phosphorylation by CaMKIIin a Ca /calmodulin (CaM)–dependent fashion (Communi et al., 1997). A further possibil-ity could eventually be that IP
3K is also inhibited by Ca -dependent PKC phosphorylation(Sim et al., 1990), however, since evidence for the existence of such inhibitory pathway is con-tradictory (Communi et al., 1995), this possibility will not be taken into further considerationin this study.Phosphorylation of IP
3K by active CaMKII (i.e. CaMKII * ) only occurs at a single threonineresidue (Communi et al., 1997, 1999), so that it can be assumed that the rate of IP
3K phos-phorylation is J ∗ K ( C ) ∝ [CaMKII ∗ ]. On the other hand, activation of CaMKII is Ca /CaM-dependent and occurs in a complex fashion because of the unique structure of this kinase, whichis composed of ∼
12 subunits, with three to four phosphorylation sites each (Kolodziej et al.,2000). Briefly, Ca increases lead to the formation of a Ca – CaM complex (CaM + ) that mayinduce phosphorylation of some of the sites of each CaMKII subunit. However, only when twoof these sites at neighboring subunits are phosphorylated, CaMKII quickly and fully activates(Hanson et al., 1994). Despite the multiple CaM + binding reactions in the inactive kinase,experiments showed that KII activation by CaM + can be approximated by a Hill equationwith unitary coefficient (De Konick and Schulman, 1998). Hence, the following kinetic reactionscheme for CaMKII phosphorylation can be assumed:4 Ca + CaM O Ω CaM + (21)KII + CaM + O b Ω b CaMKII Ω a Ω i CaMKII ∗ (22)Consider then first the binding reaction in 22. Assuming that the second step is very rapid withrespect to the first one (Thiel et al., 1988; De Konick and Schulman, 1998), the generationof CaMKII * is in equilibrium with CaMKII consumption, i.e.,[CaMKII ∗ ] ≈ Ω a Ω i [CaMKII] (23)Then, under the hypothesis of quasi-steady state for CaMKII,d[CaMKII]d t = O b [KII][CaM + ] − (Ω a + Ω b ) [CaMKII] + Ω i [CaMKII ∗ ] ≈ ∗ ] from equation 23 in the latter equation provides[CaMKII ∗ ] = K a K b [KII][CaM + ] (25)7here K a = Ω a / Ω i and K b = O b / Ω b . Defining the total kinase II concentration as[KII] T = [KII] + [CaMKII] + [CaMKII * ] and assuming it constant, equation 25 can be rewrittenas [CaMKII ∗ ] = K a [KII] T K a · H (cid:0) [CaM + ] , K m (cid:1) (26)with K m = ( K b (1 + K a )) − .The substrate concentration for the enzyme-catalyzed reaction 22 is provided by reaction 21and reads (by QSSA) [CaM + ] = [CaM] · H ( C, K ) (27)with K = O / Ω . Therefore, replacing the latter expression for [CaM + ] in equation 26, finallyprovides [CaMKII ∗ ] = K a [KII] T K a (cid:18) K m [CaM] (cid:19) − · H (cid:18) C, K K m K m + [CaM] (cid:19) (28)Defining the Ca affinity constant of IP
3K as K D = K K m / ( K m + [CaM]), the above cal-culations show that, despite its complexity, the reaction cascade underlying the activationof CaMKII can be concisely described by a Hill function of the Ca concentration ( C ) sothat [CaMKII ∗ ] ∝ H ( C, K D ). Accordingly, it is also ˆ J K ( C ) ∝ H ( C, K D ), and equation 19for IP degradation can be rewritten as J K = O K · H ( C, K D ) H ( I, K ) (29)where O K is the maximal rate of IP degradation by IP and Ca dynamics G - ChI model for IP /Ca signaling A corollary of the biological and modeling arguments exposed in the previous section is thatCa and IP signals are, generally speaking, dynamically coupled in astrocytes. This impliesthat a complete model that mimics astrocytic IP signaling must also include a descriptionof CICR. An example of such models is the so-called ChI model originally introduced byDe Pitt`a et al. (2009), which is constituted by three ODEs respectively for intracellular Ca ( C ), the IP R gating variable h and the mass-balance equation for intracellular IP lumpingterms, (16), (20) and (29), i.e.d C d t = J r ( C, h, I ) + J l ( C ) − J p ( C ) (30)d h d t = Ω h ( C, I ) ( h ∞ ( C, I ) − h ) (31)d I d t = O δ H ( C, K δ ) (1 − H ( I, κ δ )) − O K H ( C, K D ) H ( I, K ) − Ω P I (32)The above model can be extended to explicitly modeling of GPCR dynamics by a G - ChI model.To this aim, we add to the right-hand side of equation 32 the contribution of GPCR-mediatedIP synthesis given by equation 15. However, if one is interested in how GPCR kinetics evolveswith IP and Ca dynamics, then the formula for J β given by equation 2 must be used instead8f equation 15. Accordingly, the above system of equations must be completed by equation 14for astrocytic receptor activation, i.e.dΓ A d t = . . . (14)d C d t = . . . (30)d h d t = . . . (31)d I d t = O β Γ A + O δ H ( C, K δ ) (1 − H ( I, κ δ )) − O K H ( C, K D ) H ( I, K ) − Ω P I (33)Regarding the differential equations for the variables C and h above, the original formulationof the G - ChI model considered the Li-Rinzel description for CICR previously introduced inChapter 3 (Li and Rinzel, 1994). In the following, we will refer to this formulation. In practicehowever, it must be noted that any suitable model of Ca and IP R dynamics discussed inChapters 2, 3 and 16 can be adopted in lieu of the Li-Rinzel description, and accordinglydifferent models of G - ChI type may be developed, each possibly customized to study specificaspects of coupled IP and Ca signaling in astrocytes.Figure 3 illustrates some characteristics of IP and Ca dynamics reproduced by the G - ChI model. In the left panel of this figure, IP R kinetic parameters are chosen to fit, as closelyas possible, experimental data points for the steady-state open probabilities of type-2 IP Rs atfixed Ca ( solid line ) and IP concentrations ( dashed line ). In the right panel, the remainderof the parameters of the model are then set to reproduce ( solid black line ) a sample Ca traceimaged by confocal microscopy on cultured astrocytes ( gray data points ). It may be observedhow the associated IP and h oscillations predicted by the model, are almost out of phase withrespect to the Ca ones. For h , this is due to IP R kinetics, whereby an increase of cytosolicCa promotes receptor inactivation. For IP instead, this dynamics is a direct consequence ofthe Ca -dependent rate of degradation of this molecule by the IP
3K enzyme. This is a crucialaspect of intracellular IP regulation in astrocytes which is addressed more in detail below. signaling To develop the G - ChI model in Section 2, we stressed on the molecular details of the Ca dependence of the different enzymes involved in IP signaling, yet how this dependence shapesCa and IP oscillations remains to be elucidated. With this purpose, we consider in Figure 4the simple scenario of Ca oscillations triggered by repetitive stimulation of an astrocyte bypuffs of extracellular glutamate ( top three panels ), and look at the different contributions to IP production and degradation underpinning the ensuing Ca and IP dynamics ( lower panels ).With this regard, it may be noted how the total rate of IP production ( dashed line in the fourth panel from top) almost resembles the dynamics of activation of astrocyte receptors (Γ A , second panel from top) except for little bumps in correspondence of Ca pulse-like elevations( solid trace , third panel from top). Consideration of the different contributions to IP by PLC β ( orange trace ) and PLC δ ( blue trace ) reveals that, while most of IP production is driven bymGluR-mediated PLC β activation, those bumps are instead caused by PLC δ , whose activationis substantially boosted during intracellular Ca elevations.Similar arguments also hold for IP degradation ( bottom panel ). In this case, the total rateof IP degradation ( dashed line ) closely mimics IP dynamics in between Ca elevations ( greentrace , third panel from top), and is mostly contributed by Ca -independent IP-5P-mediateddegradation ( violet trace ). This scenario however changes during Ca elevations, when IP degradation, as mirrored by the dashed line which peaks in correspondence of Ca oscillations.Overall, these observations suggest that Ca -independent activity of PLC β and IP-5P vs.Ca -dependent activation of PLC δ and IP
3K account for different regimes of IP signaling.One regime corresponds to low intracellular Ca close to resting concentrations, whereby IP is mainly produced by receptor-mediated activation of PLC β against degradation by IP-5P.The other regime significantly adds to the former for sufficiently high Ca elevations, whereIP production is boosted by PLC δ , but also IP degradation is faster by IP
3K activation.The contribution to IP production and degradation by each enzyme clearly depends ontheir intracellular expression as reflected by the values of the rate constants O β , O δ , O K andΩ P in equation 33. Nonetheless, it should be noted that the existence of different regimes ofIP production and degradation is regardless of these rate values, insofar as it is set by thevalues of the Michaelis-Menten constants of the underpinning reactions, mostly K δ and K D .Remarkably, estimates of these two constants are in the range of 0 . − . µ m , that is wellwithin the range of Ca elevations expected for an astrocyte, whose average resting Ca concentration is reported to be < . µ m (Zheng et al., 2015). This assures that activation ofPLC δ and IP
3K is effective only when intracellular Ca approaches to, or increases beyond K δ and K D , as expected by the occurrence of CICR. The existence of different regimes of IP signaling shapes the time evolution of IP with respectto stimulation in a peculiar fashion. From Figure 4 ( third panel ), it may indeed be notedthat, starting from resting values, IP increases for each glutamate puff almost stepwise, till itreaches a peak (or threshold) concentration (normalized to ∼
1) that triggers CICR, therebytriggering a Ca pulse-like elevation. This Ca elevation promotes IP degradation to someconcentration between its peak and baseline values, in a sort of reset mechanism, leaving IP to increase back again to the CICR threshold until the next elevation. In between each Ca elevation, counting from the first one ending at t ≈ increasesalmost proportionally to the number of glutamate puffs, akin to an integrator of the stimulus.This may readily be proved by analytical arguments approximating, for simplicity, eachglutamate puff occurring at t k by a Dirac’s delta δ ( t − t k ), so that the external stimulus impingingon the astrocyte is modeled by Y ( t ) = G · ∆ P k δ ( t − t k ), where G · ∆ represents the glutamateconcentration delivered in the time unit per puff (i.e. its dimensions are µ m · s). Then, assumingthat in between oscillations, intracellular Ca concentration is close to basal levels, i.e. C ≈ C ,with C < ( ≪ ) K KC , K δ , K and h ≈ h ∞ , it is possible to reduce equations 14 and 33 todΓ A d t ≈ − ( O N Y ( t ) + Ω N )Γ A + O N Y ( t ) (34)d I d t ≈ − J P + J β = − Ω P I + O β Γ A (35)Using the fact that for puffs delivered at rate ν the identity R t ′′ t ′ P k δ ( t − t k )d t = ν ( t ′′ − t ′ )holds, we can solve equation 34 for Γ A obtainingΓ A ( t ) = Z t −∞ O N Y ( t ′ ) e − R tt ′ (Ω N + O N Y ( t ′′ ))d t ′′ d t ′ = Z t −∞ O N Y ( t ′ ) e − Ω N ( t − t ′ ) e − O N R tt ′ Y ( t ′′ )d t ′′ d t ′ = Z t −∞ O N Y ( t ′ ) e − (Ω N + O N G ∆ ν )( t − t ′ ) d t ′ O N Y ( t ′ ) ∗ Z Γ A ( t ) (36)where “ ∗ ” denotes the convolution operator. It is thus apparent that the fraction of acti-vated receptors Γ A ( t ) is an integral transform of the stimulus Y ( t ) by convolution with thekernel Z Γ A ( t ). Specifically, Z Γ A ( t ) may be regarded as the fraction of astrocyte receptors stim-ulated by one extracellular glutamate puff – or equivalently, by synaptic release triggered byan action potential –, and characterizes the encoding of the stimulus by the astrocyte via itsactivated receptors.The IP signal resulting from the activated receptors then evolves according to I ( t ) = Z t −∞ O β Γ A ( t ′ ) e − R tt ′ Ω P d t ′′ d t ′ = Z t −∞ O β Γ A ( t ′ ) e − Ω P ( t − t ′ ) d t ′ = O β Γ A ( t ) ∗ Z I ( t ) (37)That is the IP signal is also an integral transform of the input stimuli through the fractionof activated receptors Γ A ( t ), by convolution with the kernel Z I ( t ) = e − Ω P t . In particular,experimental evidence hints that the rate constant Ω P is often small compared to the rate ofincoming stimulation (Appendix B), so that Z I ( t ) ≈
1. In this case then, equation 37 predictsthat I ( t ) ≈ R t −∞ O β Γ A ( t ′ )d t ′ , namely that the IP signal effectively corresponds to the integralof the fraction of activated astrocyte receptors.It is also worth understanding the nature of the threshold concentration that IP must reachin order to trigger CICR. In the G - ChI model, based on the Li-Rinzel description of CICR,this threshold may be not well-defined and generally varies with the parameter choice as wellas with the shape and amplitude of the delivered stimulation (De Pitt`a et al., 2009). Considerfor example Figure 5A where the Ca response of an astrocyte ( bottom panel ) is simulatedfor different color-coded step increases of extracellular glutamate ( top panel ). It may be notedthat CICR, reflected by one or multiple Ca pulse-like increases, is triggered by glutamateconcentrations greater or equal to the orange trace . However, the IP threshold for CICR( central panel ) appears to grow with the extracellular glutamate concentration. This is reflectedby the first ’knee’ of the IP curves which reaches progressively higher values of IP concentrationas extracellular glutamate increases from orange to lime levels. At the same time, as shownby the black dashed curve in the top panel of Figure 5B, the latency for emergence of CICRsince stimulus onset ( black marks at t = 0) decreases. This can be explained by equations 34and 35, noting that, while larger glutamate concentrations promote larger receptor-mediatedIP production, this increased production is also counteracted by faster degradation by IP-5P,since this latter linearly increases with IP . Thus while larger IP production assures shorterdelays in the onset of CICR, a larger IP level must be reached to compensate for its fasterdegradation.The top panel of Figure 5B further illustrates how the latency period for CICR onset dependson the activity of the different enzymes regulating IP production and degradation. Here thedifferent colored curves were obtained repeating the simulations of Figure 5A for a 50% increaseof the activity respectively of PLC β ( orange trace ), PLC δ ( blue trace ), IP
3K ( red trace ) andIP-5P ( violet trace ). In agreement with our previous analysis, PLC β and IP-5P have the largestimpact on respectively reducing or increasing the latency period, given that they are the mainenzymes at play in IP signaling before CICR onset. The effect of an increase of IP productionby PLC δ is instead mainly significant for low glutamate concentrations, such that they couldpromote an activation of this enzyme that is comparable to that of PLC β . Conversely, IP concentrations attained to trigger CICR by different glutamate con-centrations, and its correlation with the latency for CICR onset, suggest that the mere IP concentration is not an effective indicator of the CICR threshold, rather we should considerinstead the total IP amount produced in the astrocyte cytosol during the latency period thatprecedes CICR onset, that is the integral in time of IP concentration during such period. Thisis exemplified in the bottom panel of Figure 5B where such integral is plotted as a function ofthe different latency values computed in the top panel . It may be appreciated how this integralis essentially similar for different enzyme expressions ( colored curves ) yet associated with thesame latency value.Taken together these results put emphasis on the crucial role exerted by IP signaling in thegenesis of agonist-mediated Ca elevations. In particular they suggest that the expression ofdifferent enzymes responsible of IP production and degradation, which is likely heterogeneousacross an astrocyte, could locally set different requirements for integration and encoding ofexternal stimuli by the same cell. Different mechanisms of production and degradation of IP are only one example of the possiblemany signaling pathways that could shape the nature of Ca signaling in astrocytes. There isalso compelling evidence in vitro that shape and duration of Ca oscillations could be controlledby astrocyte receptor phosphorylation by cPKCs (Codazzi et al., 2001). To better understandthis aspect of astrocyte Ca signaling, we relax the quasi steady-state approximation on cPKCphsophorylation and thus rewrite equation 8 asdΓ A d t = O N [A] n (1 − Γ A ) − (Ω N + O K P ) Γ A (38)where P denotes the cPKC * concentration at the receptors’ site. This in turn, requires to alsoconsider a description of cPKC * dynamics, whereby at least two additional equations in the G - ChI model must be included: one that takes into account P dynamics, but also a further onethat describes DAG dynamics ( D ), which is responsible for cPKC activation by Ca -dependenttranslocation of the inactive kinase to the plasma membrane (Oancea and Meyer, 1998).By QSSA, the quantity of cPKC * is conserved during receptor phosphorylation in reaction 4.In this fashion, cPKC * production and degradation are only controlled by the pair of reactions 9and 10. On the other hand, taking into account from Section 2.1 that production of cPKC * depends on the availability of the Ca -bound kinase complex cPKC ′ , we may assume at firstapproximation that reaction 9 for Ca -binding to the kinase is at equilibrium, i.e. [cPKC ′ ] =[cPKC] T H ( C, K KC ). Accordingly, we can consider cPKC * dynamics to be driven simply byreaction 10, i.e. d P d t = J KP − J KD = O KD [cPKC ′ ] · D − Ω KD P = O KD [cPKC] T H ( C, K KC ) · D − Ω KD P ≡ O KD H ( C, K KC ) · D − Ω KD P (39)where we re-defined O KD ← O KD [cPKC] T as the maximal rate of cPKC * production (in µ m s − ).To model DAG dynamics we start instead from the consideration that PLC isoenzymeshydrolyze PIP into one molecule of IP and one of DAG, so that DAG production coincides12ith that of IP (Berridge and Irvine, 1989, and see also Figure 2B). Yet, only part of this pro-duced DAG is used to activate cPKC, while the rest is mainly degraded by diacylglycerol kinases(DAGKs) into phosphatidic acid (Carrasco and M´erida, 2007) and, to a minor extent, by diacyl-glycerol lipases (DAGLs) into 2-arachidonoylglycerol (2-AG), although this latter pathway hasonly been linked to some types of metabotropic receptors in astrocytes (Bruner and Murphy,1990; Giaume et al., 1991; Walter et al., 2004). Other pathways of use of DAG are also possiblein principle, inasmuch as DAG is a key molecule in the cell’s lipid metabolism and a basiccomponent of membranes. Nonetheless there is evidence that DAG levels are strictly regu-lated within different subcellular compartments, and DAG generated by GPCR stimulation isnot usually consumed for metabolic purposes (van der Bend et al., 1994; Carrasco and M´erida,2007).DAGK activation reflects the sequence of Ca mediated translocation, DAG binding andactivation that is also required for cPKCs, so the two reactions may be thought to be character-ized by similar kinetics, yet with an important difference. Sequence analysis of DAGK α , γ – thetwo isoforms of DAGKs most likely involved in astrocytic GPCR signaling (Dominguez et al.,2013) – reveals in fact the existence of two EF-hand motifs characteristics of Ca -binding andtwo C1 domains for DAG binding (M´erida et al., 2008). In this fashion, a Hill exponent of 2instead of 1 as in equation 39 must be considered for the DAGK activating reaction, so thatDAGK-mediated DAG degradation can be modeled by J D = O D H ( C, K DC ) H ( D, K DD ) (40)Finally, to take into account other mechanisms of DAG degradation ( J A ), including but not lim-ited to DAGLs, we assume a linear degradation rate, i.e. J A = Ω D D . This is a crude approxima-tion insofar as DAGL, could also be activated in a Ca -dependent fashion (Rosenberger et al.,2007). Nonetheless, the complexity of the molecular reactions likely involved in these otherpathways of DAG degradation would require to consider additional equations in our modelwhich are beyond the scope of this chapter. The reader who is interested in these further as-pects, may refer to Cui et al. (2016) for a possible modeling approach. For the purposes of ouranalysis instead, we will consider the following equation for DAG dynamics:d D d t = J β + J δ − J KP − J D − J A = O β Γ A + O δ H ( C, K δ ) (1 − H ( I, κ δ )) + − O KD H ( C, K KC ) · D − O D H ( C, K DC ) H ( D, K DD ) − Ω D D (41)Figure 6A shows a comparison of experimental Ca and cPKC * traces with those repro-duced by the G - ChI model including equations 39 and 41. For inherent limitations of theLi-Rinzel description of the gating kinetics of IP Rs, which fails to describe these receptors’open probability for large Ca concentration (Figure 3) and predicts fast rates of receptorde-inactivation ( O /d , Table D1), the G - ChI model cannot generate Ca peaks as large asthose experimentally observed and shown here. Nonetheless we would like to emphasize howour model qualitatively matches experimental Ca -dependent cPKC * dynamics, accuratelyreproducing the phase shift between Ca and cPKC * oscillations. This phase shift is criti-cally controlled by the constant K KC for Ca binding to the kinase, along with the rates ofcPKC * production vs. degradation, i.e. O KD vs. Ω KD (equation 39), and the rate of receptorphosphorylation O K (equation 38).Figure 6B further reveals the role of these rate constants in the control of Ca oscillations.In this figure, we simulated the astrocyte response for a step increase of ∼ . µ m extracellularglutamate, starting from resting conditions, both in the absence of kinase-mediated receptor13hosphorylation ( gray trace ) and in the presence of it, for two different O K rate values ( blacktraces ). It may be noted how receptor phosphorylation by cPKC can rescue Ca oscillationsthat otherwise would vanish by saturating intracellular IP concentrations ensuing from largereceptor activation. This activation indeed is decreased by cPKC * according to equation 38,thereby regulating intracellular IP within the range of Ca oscillations. Nonetheless, asthe rate of receptor phosphorylation increases ( dash-dotted trace ), the period of oscillationsappears to slow down and oscillations even fail to emerge, if the supply of cPKC * results ina phosphorylation rate of astrocyte receptors that exceeds their agonist-mediated activation(results not shown).These considerations can be explained considering the period of Ca oscillations as a func-tion of the extracellular glutamate concentration. As shown in Figure 6C, cPKC-mediatedreceptor phosphorylation shifts ( black curves ) the range of glutamate concentrations that trig-ger Ca oscillations to higher values than those otherwise expected in the absence of it ( graycurve ). In particular, and in agreement with experimental findings (Codazzi et al., 2001), theexact value of the rate O K for receptor phopshorylation sets the entity of this shift, accountingeither for Ca oscillations of period longer than without receptor phosphorylation, or for therequirement of larger glutamate concentrations to observe such oscillations. This is respectivelyreflected by the portions of the black curves that are within the range of extracellular gluta-mate concentrations of the gray curve ), and those that instead are not. On the other hand,longer-period oscillations in the presence of receptor phosphorylation are likely to be observedas long as the rate of cPKC * activation by DAG ( O KD ) is below some critical value. A three-fold increase of this rate indeed requires glutamate concentrations beyond those needed in theabsence of receptor phosphorylation to trigger oscillations, regardless of the O K value at play( blue curves ). In this scenario in fact, the large supply of cPKC * , resulting from the high O KD value, favors phosphorylation of receptors while hindering intracellular buildup of IP totrigger CICR. This in turn requires a larger recruitment of astrocyte receptors by larger agonistconcentrations to evoke Ca oscillations. The modeling arguments introduced in this chapter overall suggest a great richness in thepossible modes whereby astrocytes could translate extracellular stimuli into intracellular Ca dynamics. These modes are brought forth by a complex network of biochemical reactions thatis exquisitely nonlinearly coupled with Ca dynamics through different second messengers,among which IP and possibly DAG could play a paramount signaling role. In particular, theregulation of different regimes of IP production and degradation by Ca in parallel with thedifferential regulation by this latter and DAG of the activities of cPKCs and DAGKs opens tothe scenario of the existence of different regimes of signal transduction that a single astrocytecould multiplex towards different intracellular targets depending on different local conditions ofneuronal activity.An interesting implication emerging from our analysis of the regulation of the period ofCa oscillations by cPKCs and DAG-related lipid signals is the possibility that these pathways,which could be crucially linked with inflammatory responses underpinning reactive astrocytosis(Brambilla et al., 1999; Griner and Kazanietz, 2007), could be found at different operationalstates, akin to what suggested for proinflammatory cytokines like TNF α (Santello and Volterra,2012). In our analysis for example, intermediate activation of cPKC activity could promoteCa oscillations at physiological rates, while an increase of it could exacerbate fast, potentiallyinflammatory Ca responses (Sofroniew and Vinters, 2010).Similar arguments also hold for IP signaling. Calcium-dependent IP production by PLC δ β (via cPKC) could modulate the rate of integration of synaptic stimuli and thus dictatethe threshold synaptic activity triggering CICR. On the other hand, the existence of differentregimes of IP degradation could be responsible for different cutoff frequencies of synapticrelease, beyond which integration of external stimuli by the cells could cease. In particular, thiscutoff frequency could be mainly set by IP-5P during low synaptic activity, possibly associatedwith low intracellular Ca levels, while be dependent on IP
3K in regimes of strong astrocyteCa activation, and thus ultimately depend on the history of activation of the astrocyte. Thefollowing chapter looks closely at some of these aspects, focusing in particular, on the role ofdifferent IP degradation regimes in the genesis and shaping of Ca oscillations.15 ppendix A Arguments of chemical kinetics A.1 The Hill equation
In biochemistry, the binding reaction of n molecules of a ligand L to a receptor macromolecule R ,i.e., R + n L k f k b RL n (42)can be mathematically described by the differential equationd[RL n ]d t = k f [R][L] n − k b [RL n ] (43)where k f , k b denote the forward (binding) and backward (unbinding) reaction rates respectively.At equilibrium, 0 = k f [R][L] n − k b [RL n ] ⇒ [RL n ] = [R][L] n K d (44)where K d = k b /k f is the dissociation constant of the binding reaction 42. Then, the fractionof bound receptor macromolecules with respect to the total receptor macromolecules can beexpressed by the Hill equation (Stryer, 1999)BoundTotal = [RL n ][R] + [RL n ] = [L] n K d [ L ] n K d + 1 = [L] n [L] n + K d = [ L ] n [ L ] n + K n . = H n ([L] , K . ) (45)where the function H n ([L] , K . ) denotes the sigmoid (Hill) function [L] n / ([L] n + K . n ), and K . = n √ K d is the receptor affinity for the ligand L , and corresponds to the ligand concentrationfor which half of the receptor macromolecules are bound (i.e. the midpoint of the H n ([L] , K . )curve). The sigmoid shape of H n ([L] , K . ) denotes saturation kinetics in the binding reac-tion 42, that is, for [L] ≫ K . almost all the receptor molecules are bound to the ligand, sothat the fraction of bound receptor molecules does not essentially change for an increase of [ L ].The coefficient n , also known as Hill coefficient , quantifies the cooperativity among multipleligand binding sites. A Hill coefficient n > positively cooperative binding , wherebyonce one ligand molecule is bound to the receptor macromolecule, the affinity of the latter forother ligand molecules increases. Conversely, a value of n < negatively cooperativebinding , namely when binding of one ligand molecule to the receptor decreases the affinity ofthe latter to bind further ligand molecules. Finally, a coefficient n = 1 denotes completely independent binding when the affinity of the receptor to ligand molecules is not affected by itsstate of occupation by the latter.For unimolecular reactions, n = 1 coincides with the number of binding sites of the receptor.For multimolecular reactions involving η > n ≤ η (Weiss, 1997). Thisfollows from the hypothesis of total allostery that is implicit in the reaction 42, whereby the Hillfunction is a very simplistic way to model cooperativity. It describes in fact the limit case whereaffinity is 0 if no ligand is bound, and infinite as soon as one receptor binds. That is, only twostates are possible: free receptor and receptor with all ligand bound. More realistic descriptionsare available in literature, such as for example the Monod–Wyman–Changeux (MWC) model,but they yield much more complex equations and more parameters (Changeux and Edelstein,2005). 16 .2 The Michaelis-Menten model of enzyme kinetics The Michaelis-Menten model of enzyme kinetics is one of the simplest and best-known modelsto describe the kinetics of enzyme-catalyzed chemical reactions. In general enzyme-catalyzedreactions involve an initial binding reaction of an enzyme E to a substrate S to form a com-plex ES. The latter is then converted into a product P and the free enzyme by a further reactionthat is mediated by the enzyme itself and can be quite complex and involve several interme-diate reactions. However, there is typically one rate-determining enzymatic step that allowsthis reaction to be modeled as a single catalytic step with an apparent rate constant k cat . Theresulting kinetic scheme thus readsE + S k f k b ES k cat P + E (46)By law of mass action, the above kinetic scheme gives rise to 4 differential equations (Stryer,1999): d[S]d t = − k f [E][S] + k b [ES] (47a)d[E]d t = − k f [E][S] + k b [ES] + k cat [ES] (47b)d[ES]d t = k f [E][S] − k b [ES] − k cat [ES] (47c)d[P]d t = k cat [ES] (47d)In the Michaelis-Menten model the enzyme is a catalyst, namely it only facilitates the reac-tion whereby S is transformed into P, hence its total concentration [E] T –– [E] + [ES] must bepreserved. This is indeed apparent by the sum of the second and the third equations above,since: d([E]+[ES])d t = d[E] T d t = 0 ⇒ [E] T = const.The system of equations 47 can be solved for the products P as a function of the concentrationof the substrate [S]. A first solution assumes instantaneous chemical equilibrium between thesubstrate S and the complex ES, i.e. d[S]d t = 0, whereby the initial binding reaction can beequivalently described by a Hill equation (Keener and Sneyd, 2008), i.e.,[ES][E] T = [S][S] + K d ⇒ [ES] = [E] T [S][S] + K d (48)Alternatively, the quasi-steady-state assumption (QSSA) that [ES] does not change on the timescale of product formation can be made, so that ddt [ES] = 0 ⇒ k f [E][S] = k b [ES] + k cat [ES](Keener and Sneyd, 2008), and k f [E][S] = k b [ES] + k cat [ES] ⇒ k f ([E] T − [ES]) [S] = k b [ES] + k cat [ES] ⇒ k f [E] T [S] = ( k f [ES][S] + k b [ES] + k cat [ES]) ⇒ [ES] = [E] T [S][S] + K M (49)where K M = ( k b + k cat ) /k f is the Michaelis-Menten constant of the reaction which quantifiesthe affinity of the enzyme to bind to the substrate.Regardless of the hypothesis made to find an expression for [ES], the rate v P of productionof P can be always written as v P = d[P]d t = k cat [ES] = k cat [E] T [S][S] + K . = v max [S][S] + K . (50)17here v max = k cat [E] T is the maximal rate of production of P in the presence of enzyme satu-ration, when all the available enzyme takes part in the reaction; and the affinity constant K . equals the dissociation constant K d of the initial binding reaction in the chemical equilibriumapproximation (equation 48), or the Michaelis-Menten constant in the QSSA (equation 49).An important corollary of the Michaelis-Menten model of enzyme kinetics is that the fractionof the total enzyme that forms the intermediate complex ES can be expressed by a Hill equationof the type [ES][E] T = [S][S] + K . = H ([ S ] , K . ) (51)and K . can be regarded as the half-saturating substrate concentration of the reaction. Simi-larly, the effective reaction rate v P (equation 51) is proportional to the maximal reaction rateby a Hill-like term H ([ S ] , K . ). Appendix B Parameter estimation
B.1 Metabotropic receptors
Rate constants O N , Ω N (equation 14) lump information on astrocytic metabotropic receptors’activation and inactivation, namely how long it takes for these receptors, once bound by theagonist, to trigger PLC β -mediated IP production and how long this latter lasts. Since IP production mediated by agonist binding with the receptors controls the initial intracellularCa surge, these two rate constants may be estimated by rise times of agonist-triggered Ca signals. With this regard, experiments reported that application of 50 µ m DHPG – a potentagonist of mGluR5 which are the main type of metabotropic glutamate receptors expressed byastrocytes (Aronica et al., 2003) –, triggers submembrane Ca signals characterized by a risetime τ r = 0 . ± .
095 s. Because mGluR5 affinity ( K . ) for DHPG is ∼ µ m (Brabet et al.,1995), that is much smaller than the applied agonist concentration, receptor saturation may beassumed in those experiments whereby the receptor activation rate by DHPG ( O DHPG ) can beexpressed as a function of τ r (Barbour, 2001), i.e. O DHPG ≈ τ r / (50 µ m ) = 0 . − . µ m − s − ,so that Ω DHPG = O DHPG K . ≈ . − .
22 s − . Corresponding rate constants for glutamatemay then be estimated assuming similar kinetics, yet with K . = K N = Ω N /O N ≈ − µ m (Daggett et al., 1995), that is 1.5–5-fold larger than K . for DHPG. Moreover, since rise timesof Ca signals triggered by non-saturating physiological stimulation are faster than in thecase of DHPG (Panatier et al., 2011), it may be assumed that O N > O DHPG . With this re-gard, for a choice of O N ≈ × O DHPG = 0 . µ m − s − , with K N = 6 µ m such that Ω N =(0 . µ m − s − )(6 µ m ) = 1 . − , a peak of extracellular glutamate concentration of 250 µ m , de-livered at t = 0 and exponentially decaying at rate Ω c = 40 s − (Clements et al., 1992), isconsistent with a peak fraction of bound receptors of ∼ .
75 within ∼
70 ms from stimulation(equation 14), which is in good agreement with experimental rise times.
B.2 IP R kinetics
We consider a steady-state receptor open probability in the form of p open ( C, I ) = H ( I, d ) ··H ( C, d )(1 − H ( C, Q )) with Q = d ( I + d ) / ( I + d ) (see Chapter 3) and choose parame-ters to fit corresponding experimental data by Ramos-Franco et al. (2000) for (i) different Ca concentrations ( ˆ C at a fixed IP level of ¯ I = 1 µ m , i.e. ˆ p ( ˆ C ); and (ii) for different IP concentra-tions ( ˆ I ) at an intracellular Ca concentration of ¯ C = 25 n m , i.e. ˆ p ( ˆ I ). To reduce the problemdimensionality while retaining essential dynamical features of IP gating kinetics we set d = d (Li and Rinzel, 1994). Accordingly, defining the vector parameter x p = ( d , d , d , O ), we min-imize the cost function c p ( x p ) = ( p open ( ˆ C, ¯ I ) − ˆ p ( ˆ C )) + ( p open ( ¯ C, ˆ I ) − ˆ p ( ˆ I )) by the Artificial18ee Colony (ABC) algorithm (Karaboga and Basturk, 2007) considering 2000 evolutions of acolony of 100 individuals.Ultrastructural analysis of astrocytes in situ revealed that the probability of ER localiza-tion in the cytoplasmic space at the soma is between ∼ ρ A ) is comprisedbetween ∼ content C T we make the consideration that the rest-ing Ca concentration in the cytosol is < . µ m (Zheng et al., 2015) and can be neglectedwith respect to the amount of Ca stored in the ER ( C ER ) (Berridge et al., 2003). Hence,with C ER ≥ µ m (Golovina and Blaustein, 1997) and a choice of ρ A ≥ .
4, it follows that C T ≈ ρ A C ER ≥ µ m . In conditions close to store depletion during oscillations (Camello et al.,2002), this latter value would also coincide with the peak Ca reached in the cytoplasm, whichis reported between < µ m and ∼ µ m (Csord`as et al., 1999; Parpura and Haydon, 2000;Kang and Othmer, 2009; Shigetomi et al., 2010).In our simulations we set ρ A = 0 . C T as far as theresulting Ca oscillations qualitatively resemble the shape of those observed in experiments.The remaining parameters for CICR, i.e. z c = (Ω C , O P ), were chosen to approximate the num-ber and period of Ca oscillations observed on average in experiments on cultured astrocytesthat were stimulated by glutamate perfusion. By “on average” we mean that we considered theaverage trace resulting from n = 5 different Ca signals generated within the same period oftime and by the same stimulus in identical experimental conditions. B.3 IP signaling Once set the CICR parameters, individual Ca traces used to obtained the above-mentioned“average trace” were used to search for z p = ( O β , O δ , O K , Ω P ), assuming random initialconditions. The ensuing parameter values were also used in Figures 4–6 although O β , O δ and O K were increased, from case to case, by a factor comprised between 1 . − O β , O δ ) or termination (bylarger O K values). B.4 cPKC and DAG signaling
Calcium-dependent cPKC-mediated phosphorylation has been documented for astrocyticmGluRs and P Y Rs (Codazzi et al., 2001; Hardy et al., 2005) and results in a reduction ofreceptor binding affinity by a factor ζ ≈ −
10 (Hardy et al., 2005), or possibly higher de-pending on the cell’s expression of cPKCs (Nakahara et al., 1997; Shinohara et al., 2011). Sinceexperiments showed that cPKC is robustly activated only when Ca increases beyond half ofthe peak concentration reached during oscillations (Codazzi et al., 2001) then, considering peakCa values of ∼ − µ m (Shigetomi et al., 2010) allows estimating Ca affinity of cPKCin the range of K KC ≤ . − . µ m which indeed comprises the value of ∼
700 n m predictedexperimentally (Mosior and Epand, 1994). Of the same order of magnitude also is the Ca affinity reported for DAGK, i.e. K DC ≈ . − . µ m (Sakane et al., 1991; Yamada et al., 1997).Reported values of DAG affinities for cPKC and DAGK may considerably differ. Micellar as-says of cPKCs activity, suggests values of K KD as low as 4.6–13 . m (Ananthanarayanan et al.,2003), whereas studies on purified DAGK suggest a substrate affinity for this kinase of K DD ≈ µ m (Kanoh et al., 1983). The differences in experimental setups and the possibility that theactivity of these kinases could be widely regulated by different DAG pools make these estimateof scarce utility for our model, where the DAG concentration is of the same order of magnitude19f IP one. With this regard we choose to set these affinities to 0 . µ m which corresponded inour simulations to the average intracellular DAG concentration during Ca oscillations.The remaining parameters, namely z k = ( O KD , O K , Ω D , O D , Ω D ) were arbitrarily chosenconsidering two constrains: (i) DAG concentration for damped Ca oscillations must stabilizeto a constant value; and (ii) the down phase of cPKC * oscillations must follow that of Ca ones as suggested by experimental observations by Codazzi et al. (2001). Appendix C Software
The Python file figures.py used to generate the figures of this chapter can be downloadedfrom the online book repository at https://github.com/mdepitta/comp-glia-book . Thesoftware for this chapter is organized in two folders. The data folder contains data to fit the G - ChI model. WebPlotDigitizer 4.0 ( https://automeris.io/WebPlotDigitizer ) was usedto extract experimental data by Ramos-Franco et al. (2000, Figures 6 and 7) and Codazzi et al.(2001, Figure 5). The Jupyter notebook file data_loader.ipynb found in this folder containsthe code to load and clean experimental data used in the simulations.The code folder contains instead all the routines (including figures.py ) used for the simu-lations of this chapter. The two files astrocyte_models.h and astrocyte_models.cpp contains thecore G - ChI model implementation in C/C++11, while the class
Astrocyte in astrocyte_models.py provides the Python interface to simulate the G - ChI model. The modelwas integrated by a variable-coefficient linear multistep Adams method in Nordsieck form whichproved robust to correctly solve stiff problems rising from different parameter choices (Skeel,1986). Model fitting is provided by gchi_fit.py and relies on the PyGMO 2.6 optimizationpackage ( https://github.com/esa/pagmo2.git ).The library gchi_bifurcation.py provides routines to estimate the period and range of Ca oscillation as in Figures 6. These routines use numerical continuation of the extended G - ChI model by the Python module PyDSTool 0.92 (Clewley, 2012, https://github.com/robclewley/pydstool ).20 ppendix D Model parameters used in simulations
Table D1.
Model parameters used in the simulations, unless differently specified in figurecaptions.Symbol Description Value Units
Astrocyte receptors Ω N Rate of receptor de-activation 1.8 s − O N Rate of agonist-mediated receptor activation 0.3 µ m − s − n Agonist binding cooperativity 1 – IP R kinetics d IP binding affinity 0.1 µ m O Inactivating Ca binding rate 0.325 µ m − s − d Inactivating Ca binding affinity 4.5 µ m d IP binding affinity (with Ca inactivation) 0.1 µ m d Activating Ca binding affinity 0.05 µ m Ca fluxes C T Total ER Ca content 5 µ m ρ A ER-to-cytoplasm volume ratio 0.5 –Ω C Maximal Ca release rate by IP Rs 7.759 s − Ω L Ca leak rate 0.1 s − O P Maximal Ca uptake rate 5.499 µ m s − K P Ca affinity of SERCA pumps 0.1 µ m IP production O β Maximal rate of IP production by PLC β µ m s − O δ Maximal rate of IP production by PLC δ µ m s − K δ Ca affinity of PLC δ µ m κ δ Inhibiting IP affinity of PLC δ µ m IP degradation Ω P Rate of IP degradation by IP-5P 0.86 s − O K Maximal rate of IP degradation by IP
3K 0.86 µ m s − K D Ca affinity of IP
3K 0.5 µ m K K IP affinity of IP
3K 1.0 µ m DAG dynamics Ω D Unspecific rate of degradation 0.26 s − O D Rate of degradation by DAGK 0.45 µ m s − K DC DAGK affinity for Ca µ m K DD DAGK affinity for DAG 0.1 µ m cPKC signaling O KD Rate of cPKC * production 0.28 µ m s − Ω KD Rate of cPKC * deactivation 0.33 s − K KC Ca affinity of PKC 0.5 µ m O K Rate of receptor phosphorylation 1.0 µ m − s − B Figure 1. IP production. A Hydrolysis of the membrane lipid phosphatidylinositol 4,5-bisphosphate (PIP ) by PLC β and PLC δ isoenzymes produces IP and diacylglycerol (DAG).The contribution of PLC β to IP production depends on agonist binding to astrocyte G protein-coupled receptors (GPCRs). This production pathway is inhibited via receptor phosphorylationby Ca -dependent activation of conventional protein kinases C (cPKCs). Blue: promotingpathway; red : inhibitory pathway. 22 B Figure 2. IP and DAG degradation. A Degradation of IP occurs by phosphorylation intoinositol 1,3,4,5-tetrakisphosphate (IP ) by IP
3K and dephosphorylation into lower inositolphosphates by IP-5P. Both pathways are regulated by Ca : IP
3K activity is stimulated byphosphorylation by Ca /calmodulin-dependent protein kinase II (CaMKII), whereas IP-5P isinhibited thereby. Moreover IP andDAG-dependent cPKC-mediated phosphorylation, while IP-5P could also be inhibited by IP .For the sake of simplicity, IP-5P dependence on Ca and IP along with IP
3K dependenceon cPKC are not taken into consideration in this study ( dashed pathways ). B DAG is mainlydegraded into phosphatidic acid (PA) by DAG kinases (DAGK) in a Ca -dependent fashion,and to a minor extent, into 2-arachidonoylglycerol (2-AG) by DAG lipases (DAGL). In turn 2-AG is hydrolized by monoacylglycerol lipase (MAGL) into arachidonic acid (AA). 2-AG and AAmay promote activity of DAGK and cPKC * ( orange patwhays ) although this scenario is nottaken into consideration here. Colors of other pathways as in Figure 1.23 .01 0.1 1.0 10.0 100.0 Ca , IP (μM) I P R o p e n p r o b a b ili t y Ca =25 nMIP =1 μM Time (s) C h I v a r i a b l e s ( n . u . ) ChI
Figure 3. G - ChI model . ( left panel ) Fit of IP Rs kinetic parameters on experimentaldata of steady-state open probabilities of type-2 IP Rs by Ramos-Franco et al. (2000). Inthis example, and through all this chapter, we consider the Li-Rinzel description for CICR.This choice allows a reasonable fit ( solid and dashed lines ) of the receptors’ open probabilityas function of either intracellular IP ( N ) or intracellular Ca ( ). The only exception isfor Ca concentrations > µ m for which the open probability predicted by the Li-Rinzelmodel ( solid line ) vanishes much more quickly than experimental values. ( right panel ) SampleCa ( C ), IP ( I ) and h traces ensuing from a simulation of the G - ChI model to reproduceexperimental Ca oscillations in cultured astrocytes ( gray data points ) triggered by applicationof > µ m glutamate. Experimental data courtesy of Nitzan Herzog (University of Nottingham).A saturating glutamate concentration (i.e. Γ A = 1) was assumed with initial conditions C (0) =0 . µ m , h (0) = 0 .
972 and I (0) = 0 . µ m . Simulated Ca and IP traces are reportedin normalized units with respect to minimum values of C = 0 . µ m and I = 0 . µ m andpeak values of ˆ C = 1 . µ m and ˆ I = 0 . µ m . Model parameters as in Table D1 except for O β = 0 . µ m s − and O K = 0 . µ m s − . 24 l u ( μ M ) A s t . R e c . Γ A C a + , I P ( . u . ) CI I P p r o d . ( μ M / s ) J β J δ Time (s) I P d e g r . ( μ M / s ) J J Figure 4.
Coexistence of different regimes of IP signaling. From top to bottom: ( firstpanel ) Repetitive stimulation of an astrocyte by puffs of glutamate (8 µ m , rectangular pulsesat rate 0 .
33 Hz and 15% duty cycle); ( second panel ) fraction of activated astrocytic receptors;( third panel ) ensuing Ca ( C ) and IP ( I ) traces (normalized with respect to their maximumexcursion: C = 40 n m , I = 50 n m , ˆ C = 0 . µ m , ˆ I = 0 . µ m ); ( fourth panel ) total rate ofIP production ( dashed line ) and contributions to it by PLC β ( J β ) and PLC δ ( J δ ); ( bottompanel ) total rate of IP degradation ( dashed line ) resulting from the combination of degradationby IP-5P ( J P ) and IP
3K ( J K ). Besides Ca pulsed-oscillations, IP is mainly regulated byPLC β ( orange trace ) and IP-5P ( violet trace ), and its concentration tends to increase in anintegrative fashion with the number of glutamate puffs. During Ca elevations instead, activityof PLC δ ( blue trace ) and IP
3K ( red trace ) become significant, with this latter responsible for asharp drop of intracellular IP . Model parameters as in Table D1 except for C T = 10 µ m , O P =10 µ m s − and O δ = 0 . µ m s − . 25 G l u ( μ M ) I P ( μ M ) Time (s) C a + ( μ M ) B Glu (μM) L a t e n c y ( s ) referenceO β (×1.5)O δ (×1.5)O (×1.5)Ω (×1.5) Latency (s) I P T i m e I n t e g r a l ( m M ⋅ s ) Figure 5.
Threshold for CICR. A ( top panel ) Step increases of extracellular glutamate ( colorcoded ) and resulting IP ( central panel ) and Ca ˜dynamics ( bottom panel ) in a G - ChI astrocytemodel.
Black marks at t = 0 denote stimulus onset. B ( top panel ) Latency for the onset ofCICR as a function of the applied glutamate concentration for the Ca traces in A ( blackdashed curve ), as well as for 50% increases in the rate of PLC β ( O β ), PLC δ ( O δ ), IP
3K ( O K )and IP-5P (Ω P ) respectively. Emergence of CICR was detected for d C d t ≥ . µ m / s. ( bottompanel ) Integral of IP concentration as a function of the latency values computed in the toppanel. This integral is a better estimator of CICR threshold than the sole IP concentration.Model parameters as Figure 4. 26 E x p e r i m e n t a l T r a c e s ( n . u . ) Ca cPKC * Time (s) S i m u l a t e d T r a c e s ( n . u . ) B C a + ( μ M ) D A G ( μ M ) Time (s) c P K C * ( n M ) O K =0O K =1.0 μM −1 s −1 O K =3.0 μM −1 s −1 C Glu (μM) O s c ill a t i o n P e r i o d ( s ) O KD =0O KD =0.28 μMs −1 O KD =0.84 μMs −1 Figure 6.
Regulation of Ca oscillations by cPKC. A ( top panel ) Comparison between ex-perimental traces for Ca ( black ) and cPKC * ( red ) originally recorded in cultured astrocytesby Codazzi et al. (2001) and simulations ( bottom panel ). Despite quantitive differences in theshape and period of oscillations, the model can reproduce the essential correlation and phaseshift between Ca and cPKC * dynamics observed in experiments. Ca and cPKC * oscil-lations were triggered assuming an extracellular glutamate concentration of 1 . µ m , and werenormalized according to their maximum excursion: C = 0 . µ m , P = 48 n m , ˆ C = 0 . µ m andˆ P = 65 n m . B DAG and cPKC * dynamics associated with two different rates of receptor phos-phorylation by cPKC ( O K , black traces ) in response to a step increase of extracellular glutamate(1 . µ m at t = 0). In the absence of receptor phosphorylation ( gray traces ), Ca oscillationswould vanish due to saturating intracellular IP levels ensued from large receptor activation. C Period of Ca oscillations as a function of extracellular glutamate concentration. Receptorphosphorylation by cPKC critically controls the oscillatory range ( black and blue curves ) withrespect to the scenario without cPKC activation ( gray curve ). Higher glutamate concentrationsare required to trigger oscillations for larger rates of DAG-dependent cPKC activation ( O KD ).Parameters as in Table D1 except for Ω C = 6 .
207 s − , Ω L = 0 .
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