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Nonlinear gapjunctions enable long-distance propagation of pulsating calcium waves in astrocyte networks
Mati Goldberg, Maurizio De Pitt, Vladislav Volman, Hugues Berry and Eshel Ben-Jacob
SUPPLEMENTARY INFORMATION
The ChI model for intracellular Ca2+ dynamics
Intracellular calcium dynamics in cells are the result of an intricate signaling pathway that mostly rests on Ca2+ exchange between the cytoplasm and internal stores, such as the endoplasmic reticulum (ER) or mitochondria [1]. In astrocytes, Ca2+ exchange with the ER plays the prominent role [2] and the concentration of cytoplasmic Ca2+ depends on three distinct fluxes (illustrated in Figure1a).
(1)Transfer of Ca2+ from the cytoplasm to the ER lumen is mediated by an active pump, the (Sarco-)Endoplasmic-reticulum Ca2+ ATPase (SERCA). SERCA contributes to the maintenance of higher Ca2+ concentrations in the ER than in the cytoplasm.
(2)This concentration gradient drives a passive Ca2+ leak from the ER to the cytoplasm.
(3)Specific Ca2+ transporters, called inositoltriphosphate receptor-channels (IP3Rs, for short), contribute a third flux, transporting molecules from the ER to the cytoplasm. The regulation of IP3Rs is remarkable: their activity is upregulated by a small molecule, inositoltriphosphate (IP3) and by small-to-intermediate cytoplasmic Ca2+ levels. However, large Ca2+ concentrations inhibit IP3R activity.
Therefore, a small increase in cytoplasmic Ca2+ levels activates IP3Rs, which leads to increased Ca2+ transport from the ER, ultimately increases cytoplasmic Ca2+. Because of this positive feedback, Ca2+ transport through IP3Rs is usually called Ca2+-induced Ca2+release (CICR). At some point however, the concentration of Ca2+ within the cytoplasm becomes large enough to inhibit IP3R activity and the surge of Ca2+ through IP3R terminates. These three transport mechanisms are classically captured by a mathematical model due to Li and Rinzel [3-5], given by eq.(1) and eq.(2) in the article. As is obvious from these equations, the Li-Rinzel (LR) model has two dynamical variables: C, the free cytosolic Ca2+ concentration and h, the fraction of inactive (closed) IP3R channels. The concentration of IP3 in the cytoplasm (IP3) instead is a parameter. To simulate this system, one must specify the mathematical formulation of the three fluxes Jpump, Jleak and Jchan as well as those of the steady fraction of closed IP3Rs, h" and of the channel open/close transition time constant, th. In the present work, we have used the classical formulations originally introduced by Li and Rinzel [3-5].
As mentioned above, one feature of the Li-Rinzel model is that it treats the concentration of IP3 as a mere parameter, i.e. IP3 in equations (1-2) is a constant value in time and does not change when the dynamical variables C and h vary over time. Yet, inspection of the intracellular Ca2+ signaling pathway (Figure1b) immediately shows that IP3 concentration in the cytosol is coupled to Ca2+ dynamics, and therefore is expected to change with C. In particular, IP3 formation and degradation are both regulated by cytoplasmic Ca2+.
We recently presented a model called the ChI model, that basically extends the Li-Rinzel model but includes IP3 as a third dynamical variable [5]. To this aim, we had to simplify the complexity of the IP3 signaling pathway (Figure1c,d). This allowed us to derive a tractable yet realistic model. For clarity, we briefly describe the ChI model below. Further details and analyses can be found in [5].
In astrocytes, endogenous IP3 dynamics (Figures 1c,d) mainly result from three enzymatic processes [6-8]. Endogeneous IP3 is produced from a membrane lipid called phospatidylinositolbiphosphate (PIP2) in a reaction catalyzed by phospholipaseC (PLC). It can be degraded through phosphorylation (catalyzed by IP33-kinase, IP3-3K) or through dephosphorylation (catalyzed by inositol polyphosphate 5-phosphatase, IP-5P). Experimentally, PLC is activated by cytoplasmic Ca2+ in a cooperative fashion with a Hill exponent close to 2. But PLCd is also inhibited by IP3 concentrations larger than circa 1M. We thus expressed IP3 production through PLCd as:
EMBED Equation.3 (S1)
where is the maximal rate of IP3 production by PLC, and we define generic Hill functions as EMBED Equation.3 . Dephosphorylation of IP3 by IP-5P displays pure Michaelis-Menten kinetics, but its Michaelis constant (H"10M) is higher than physiological IP3 levels. It can thus be expressed as a linear reaction term:
EMBED Equation.3 (S2)
As evidenced in Figure1d, the regulation of IP33-kinase by cytoplasmic Ca2+ is a complicated mechanism and includes interactions between Ca2+, calmodulin and CaMKII. Using the algebraic properties of Hill functions however, we have been able to lump all this complexity into the product of a 4th-order Hill function of Ca2+ with a first-order Hill function of IP3 (see [5] for further explanation):
EMBED Equation.3 (S3)
Taken together, the IP3 concentration in the ChI model is a third dynamical variable, whose dynamics are given by:
EMBED Equation.3 (S4)
where vbias defines an external input of IP3 coming e.g. from a neighboring astrocyte cell coupled through gap junction. Note that equation(S4) yields to equation(3) in the article when vbias=0. Equations (1-2) and (S1-S4) together define the ChI model of intracellular Ca2+ dynamics.
Bifurcation analysis of the ChI model and encoding modes
a) Isolated cell
We start our analysis with a description of the dynamics of the uncoupled case, i.e.the ChImodel equations(1-2) and (S1-S4). FigureS1 shows bifurcation diagrams when the input bias of IP3 varies in equation (S4). For either small or large biases, the IP3 production term in equation (S4) is balanced by the two degradation terms. This gives rise to two stable steady-(time independent) states (black thick lines in FiguresS1a,b). Further analysis of equations(1-2,S1-S4) shows that between the two steady-state regimes, spontaneous oscillations of the three variables (C, h and IP3) arise. The evolution of the envelop of these oscillations (minimal and maximal values during the cycle) with bias is shown in 3D plots as thin gray lines (Figures S1a,b) or in 2D projections as full thick lines (Figures S1c,d).
The general scenario indicated above (i.e. low steady state then oscillations then high steady state as vbias increases) is qualitatively conserved when the cell parameters vary in realistic ranges. However, a key observation for the present work is that, depending on the value of some cell physiological parameters (e.g. the affinity of SERCA pumps), the nature of the obtained oscillatory behavior changes. The major change concerns the type of bifurcation that is associated with the birth of the oscillatory behavior. Oscillations can arise through a Hopf bifurcation (FigureS1a) or through a Saddle-Node-on-Invariant-Circle (SNIC) bifurcation (FiguresS1b). Now, the nature of the bifurcation decides how a change in the incoming IP3 bias is encoded in the Ca2+ oscillations. In fact one can classify these encoding behaviors by three classes.
(1)The amplitude of Ca2+ oscillations more than doubles across the whole oscillatory range while their frequency stays almost constant. We refer to this behavior as the amplitude-modulation (AM) encoding mode.
(2)The frequency of Ca2+ oscillations more than doubles across the whole oscillatory range while their amplitude stays almost constant. We refer to this behavior as the frequency-modulation (FM) encoding mode.
(3)Both the amplitude and the frequency of Ca2+ oscillations change by more than a factor of 2 across the whole oscillatory range. We refer to this mode as the mixed amplitude-frequency-modulation (AFM) encoding mode.
Note that the waveform of the oscillations is also affected by the difference of the bifurcation type: FM-type oscillations usually display pulse-like waveforms, whereas AM-type ones are closer to sinusoidal [5].
In FigureS1 we illustrate the encoding modes displayed by the ChI model. The left-hand column (Figures S1a,c,e) illustrates a typical AFM behavior: increasing the IP3 input bias yields an increase of the oscillation amplitude of IP3 and C oscillations (FigureS1c) but also a marked decrease of their period (FigureS1e). The situation is more complex for the parameters used in the right-hand column: here the period varies dramatically but the amplitude of intracellular Ca2+ oscillations is close to constant (though the amplitude of IP3 oscillations varies). This corresponds to a FMregime (at least for C). Furthermore, as Ca2+ transport through the SERCA pumps is larger for FM conditions [9], the range of IP3 bias for which oscillations are predicted, is much narrower in theAFM regime than in theFM one.
b) Coupled cell network
We now consider a linear chain made of NChIcells coupled to their two nearest neighbors by gap junctions as defined by equations(7-13). In the case of nonlinear coupling at least, one could expect that the dynamics of the cells would be different from the behavior predicted in the uncoupled (isolated) case studied above. To evaluate these changes, we first computed bifurcation diagrams for the coupled system (while FigureS1 shows bifurcation diagrams for isolated cells).
Unfortunately, deriving such bifurcation diagrams is arduous and uncertain from a numerical point of view because the resulting system is strongly nonlinear. As a consequence, the corresponding diagrams contain imperfectly resolved parts (see FigureS2 caption). FigureS2 shows bifurcation diagrams for N=7 FM-encoding cells coupled by sigmoid-like gap junctions with reflective boundaries. We apply the IP3 stimulus on the central cell in the chain (namely Cell4 in FigureS2). In this case, we can consider only half of the chain and plot bifurcation diagrams for cells4 to7 only, since for symmetry reasons cells3 to1 display the same behavior as cells5 to7.
The bifurcation diagram of the central cell is qualitatively similar to the uncoupled case (compare Cell4 in FigureS2 with FiguresS1b, d). In particular, oscillations still arise through a SNIC bifurcation (at IP3biasH"0.72M). On the other hand, the bifurcation diagram beyond the SNIC displays multiple period-doubling cascades as well as different co-existing stable oscillatory regimes. As a consequence, the oscillatory regime in the central cell rapidly becomes extremely complex. The bifurcation diagrams for the other cells are also globally similar to that of the central (stimulated) cell, except that the fixed-point branches tend to get more horizontal when the distance from the stimulated cell increases. Furthermore, qualitatively similar diagrams are also obtained for different boundary conditions as well as for different stimulus locations (results not shown).
In general, these diagrams show FM pulse-like oscillatory regimes at low IP3bias values, which can turn into complex oscillations for larger IP3bias values. Yet because stable oscillation regimes can coexist in the bifurcation diagrams with stable fixed points, it cannot be predicted from these diagrams whether an IP3 input arriving to a given cell from its neighbor will trigger pulse-like oscillations or not, i.e.whether it will switch the system from the fixed point to the oscillatory regime. Accordingly, we used extensive numerical simulations to investigate under what conditions one could observe intercellular propagation of Ca2+ waves along the astrocyte chain.
In agreement with previous studies (see [10] for a review), IP3-triggered CICR indeed allows intercellular Ca2+ wave propagation in our modeling framework, as shown in FigureS3. Here the simple case of N=5 FMChIastrocytes coupled through sigmoid gap junctions is considered and the first cell (cellA1) is stimulated from t=10s to t=30s by a constant bias of IP3bias=0.8M. In agreement with the bifurcation diagrams shown in FigureS2, pulse-like Ca2+ oscillations are observed. Note that the 20-second long stimulation used here is shorter than the predicted period of oscillations at such IP3bias value. Therefore, only a single pulse is observed. In general when a large-enough IP3 pulse arrives to a given cell, either by direct external stimulation or from a neighbor coupled cell, CICR first generates a neat pulse-like Ca2+ response (FigureS3, black curves), which predates a pulse-like increase of IP3 (FigureS3,dashed blue curves). Part of this IP3 then diffuses to the neighbor cells in the form of a new pulse. Therefore, as long as the newly formed IP3 pulses are large enough, IP3-triggered CICR is preserved and a propagating pulsating Ca2+ wave, mediated by IP3 diffusion, is observed.
On the contrary, lack of IP3 stops wave propagation, as it happens for example in FigureS3 for the IP3 pulse that leaves cellA3 for cellA4. This pulse is indeed too small to trigger CICR in cellA4, so that Ca2+ wave propagation ceases beyond cellA3. Notably, the distance (in terms of number of cells) that a Ca2+ wave can travel along the chain strongly depends both on the nature of the gap-junction coupling and on cell type (AFM or FM) as is shown in the article.
References
Berridge MJ, Lipp P, Bootman MD (2000) The versatility and universality of calcium signalling. Nat Rev Mol Cell Biol 1: 1121.
Agulhon C, Petravicz J, McMullen AB, Sweger EJ, Minton SK, et al. (2008) What is the role of astrocyte calcium in neurophysiology? Neuron 59: 932946.
Li Y, Rinzel J (1994) Equations for InsP3 receptormediated [Ca2+]i oscillations derived from a detailed kinetic model: A HodgkinHuxley like formalism. J Theor Biol 166: 461473.
De Pitt M, Goldberg M, Volman V, Berry H, BenJacob E (2009) Glutamatedependent intracellular calcium and IP3 oscillating and pulsating dynamics in astrocytes. J Biol Phys 35: 383411.
De Pitt M, Volman V, Levine H, Pioggia G, De Rossi D, et al. (2008) Coexistence of amplitude and frequency modulations in intracellular calcium dynamics. Phys Rev E 77:030903(R).
Rebecchi MJ, Pentyala SN (2000) Structure, function, and control of phosphoinositidespecific phospholipase C. Physiol Rev 80: 12911335.
Sims CE, Allbritton NL (1998) Metabolism of inositol 1,4,5trisphosphate and inositol 1,3,4,5tetrakisphosphate by the oocytes of Xenopus laevis. J Biol Chem 273: 40524058.
Irvine RF, Letcher AJ, Heslop JP, Berridge MJ (1986) The inositol tris/tetrakisphopshate pathway demonstration of Ins(1,4,5)P33kinase activity in animal tissues. Nature 320: 631634.
De Pitt M, Volman V, Levine H, BenJacob E (2009b) Multimodal encoding in a simplified model of intracellular calcium signaling. Cogn Proc 10: 5570.
Falcke M (2004) Reading the patterns in living cells the physics of Ca2+ signaling. Adv Phys 53: 255440.
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