G Protein-Coupled Receptor-Mediated Calcium Signaling in Astrocytes

Abstract

Astrocytes express a large variety of G protein-coupled receptors (GPCRs) which mediate the transduction of extracellular signals into intracellular calcium responses. This transduction is provided by a complex network of biochemical reactions which mobilizes a wealth of possible calcium-mobilizing second messenger molecules. Inositol 1,4,5-trisphosphate is probably the best known of these molecules whose enzymes for its production and degradation are nonetheless calcium-dependent. We present a biophysical modeling approach based on the assumption of Michaelis–Menten enzyme kinetics, to effectively describe GPCR-mediated astrocytic calcium signals. Our model is then used to study different mechanisms at play in stimulus encoding by shape and frequency of calcium oscillations in astrocytes.

Eshel Ben-Jacob: Deceased June 5, 2015.

Appendices

Appendix 1 Arguments of Chemical Kinetics

1.1 Appendix 1.1 The Hill Equation

In biochemistry, the binding reaction of n molecules of a ligand L to a receptor macromolecule R, i.e.,

$$\begin{aligned} \mathrm{{R + }}\,n\,\mathrm{{L}} \mathop \rightleftharpoons ^{k_{f}}_{k_{b}} \mathrm{{RL}}_n \end{aligned}$$

(5.42)

can be mathematically described by the differential equation

$$\begin{aligned} \frac{\mathrm {d}\mathrm{{[RL}}_n]}{\mathrm {d}t}=k_{f}\mathrm{{[R][L]}}^n-k_{b}\mathrm{{[RL}}_n] \end{aligned}$$

(5.43)

where $k_{f}$, $k_{b}$ denote the forward (binding) and backward (unbinding) reaction rates, respectively. At equilibrium,

$$\begin{aligned} 0 = k_{f}\mathrm{{[R][L]}}^n-k_{b}\mathrm{{[RL}}_n] \Rightarrow \mathrm{{[RL}}_n]=\frac{\mathrm{{[R][L]}}^n}{K_d} \end{aligned}$$

(5.44)

where $K_{d}=k_{b}/k_{f}$ is the dissociation constant of the binding reaction 5.42. Then, the fraction of bound receptor macromolecules with respect to the total receptor macromolecules can be expressed by the Hill equation (Stryer 1999)

(5.45)

where the function $\mathcal {H}_{n}\left( \mathrm{{[L]}},K_{0.5}\right)$ denotes the sigmoid (Hill) function $\mathrm{{[L]}}^{n}\mathrm{{ / ([L]}}^{n}\mathrm{{ + K_{0.5}}}^{n})$, and $K_{0.5} = \root n \of {K_{d}}$ is the receptor affinity for the ligand L and corresponds to the ligand concentration for which half of the receptor macromolecules are bound (i.e., the midpoint of the $\mathcal {H}_{n}\left( \mathrm{{[L]}},K_{0.5}\right)$ curve). The sigmoid shape of $\mathcal {H}_{n}\left( \mathrm{{[L]}},K_{0.5}\right)$ denotes saturation kinetics in the binding reaction 5.42, that is, for $\mathrm{{[L]}}\gg K_{0.5}$ almost all the receptor molecules are bound to the ligand, so that the fraction of bound receptor molecules does not essentially change for an increase of $\mathrm {[L]}$.

The coefficient n, also known as Hill coefficient, quantifies the cooperativity among multiple ligand-binding sites. A Hill coefficient $n>1$ denotes positively cooperative binding, whereby once one ligand molecule is bound to the receptor macromolecule, the affinity of the latter for other ligand molecules increases. Conversely, a value of $n<1$ denotes negatively cooperative binding, namely when binding of one ligand molecule to the receptor decreases the affinity of the 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 its state of occupation by the latter.

For unimolecular reactions, $n=1$ coincides with the number of binding sites of the receptor. For multimolecular reactions involving $\eta > 1$ ligand molecules instead, the Hill coefficient, in general, only loosely estimates the number of binding sites, being $n\le \eta$ (Weiss 1997). This follows from the hypothesis of total allostery that is implicit in the reaction 5.42, whereby the Hill function is a very simplistic way to model cooperativity. It describes in fact the limit case where affinity is 0 if no ligand is bound and infinite as soon as one receptor binds. That is, only two states are possible: free receptor and receptor with all ligand bound. More realistic descriptions are available in literature, such as the Monod–Wyman–Changeux (MWC) model, but they yield much more complex equations and more parameters (Changeux and Edelstein 2005).

1.2 Appendix 1.2 The Michaelis–Menten Model of Enzyme Kinetics

The Michaelis–Menten model of enzyme kinetics is one of the simplest and best-known models to describe the kinetics of enzyme-catalyzed chemical reactions. In general enzyme-catalyzed reactions involve an initial binding reaction of an enzyme $\mathrm{{E}}$ to a substrate $\mathrm{{S}}$ to form a complex $\mathrm{{ES}}$. The latter is then converted into a product $\mathrm{{P}}$ and the free enzyme by a further reaction that is mediated by the enzyme itself and can be quite complex and involve several intermediate reactions. However, there is typically one rate-determining enzymatic step that allows this reaction to be modeled as a single catalytic step with an apparent rate constant $k_{\mathrm {cat}}$. The resulting kinetic scheme thus reads

$$\begin{aligned} \mathrm{{E + S}} \mathop \rightleftharpoons ^{k_{f}}_{k_{b}} \mathrm{{ES}} \xrightarrow {k_{\mathrm {cat}}} \mathrm{{P + E}} \end{aligned}$$

(5.46)

By law of mass action, the above kinetic scheme gives rise to four differential equations (Stryer 1999):

$$\begin{aligned} \frac{\mathrm {d}\mathrm{{[S]}}}{\mathrm {d}t}&= -k_{f}\mathrm{{[E][S]}} + k_{b}\mathrm{{[ES]}}\end{aligned}$$

(5.47a)

$$\begin{aligned} \frac{\mathrm {d}\mathrm{{[E]}}}{\mathrm {d}t}&= -k_{f}\mathrm{{[E][S]}} + k_{b}\mathrm{{[ES]}} + k_{\mathrm {cat}}\mathrm{{[ES]}} \end{aligned}$$

(5.47b)

$$\begin{aligned} \frac{\mathrm {d}\mathrm{{[ES]}}}{\mathrm {d}t}&= k_{f}\mathrm{{[E][S]}} - k_{b}\mathrm{{[ES]}} - k_{\mathrm {cat}}\mathrm{{[ES]}} \end{aligned}$$

(5.47c)

$$\begin{aligned} \frac{\mathrm {d}\mathrm{{[P]}}}{\mathrm {d}t}&= k_{\mathrm {cat}}\mathrm{{[ES]}} \end{aligned}$$

(5.47d)

In the Michaelis–Menten model, the enzyme is a catalyst, namely it only facilitates the reaction whereby $\mathrm{{S}}$ is transformed into $\mathrm{{P}}$, hence its total concentration $\mathrm{{[E]_T=[E] + [ES]}}$ must be preserved. This is indeed apparent by the sum of the second and the third equations above, since: $\frac{\mathrm {d}(\mathrm{{[E] + [ES]}})}{\mathrm {d}t}=\frac{\mathrm {d}\mathrm{{[E]_T}}}{\mathrm {d}t}=0\Rightarrow \mathrm{{[E]_T}} = \text {const}$.

The system of Eq. 5.47 can be solved for the products $\mathrm{{P}}$ as a function of the concentration of the substrate $\mathrm{{[S]}}$. A first solution assumes instantaneous chemical equilibrium between the substrate $\mathrm{{S}}$ and the complex $\mathrm{{ES}}$, i.e., $\frac{\mathrm {d}\mathrm{{[S]}}}{\mathrm {d}t}=0$, whereby the initial binding reaction can be equivalently described by a Hill equation (Keener and Sneyd 2008), i.e.,

$$\begin{aligned} \frac{\mathrm{{[ES]}}}{\mathrm{{[E]_T}}} = \frac{\mathrm{{[S]}}}{\mathrm{{[S]}} + K_{d}} \Rightarrow \mathrm{{[ES]}} = \frac{\mathrm{{[E]_T [S]}}}{\mathrm{{[S]}} + K_{d}} \end{aligned}$$

(5.48)

Alternatively, the quasi steady-state assumption (QSSA) that $\mathrm{{[ES]}}$ does not change on the timescale of product formation can be made, so that $\frac{d}{dt}\mathrm{{[ES]}} = 0 \Rightarrow k_{f}\mathrm{{[E][S]}} = k_{b}\mathrm{{[ES]}} + k_{\mathrm {cat}}\mathrm{{[ES]}}$ (Keener and Sneyd 2008), and

$$\begin{aligned} k_{f}\mathrm{{[E][S]}} = k_{b}\mathrm{{[ES]}} + k_{\mathrm {cat}}\mathrm{{[ES]}}&\Rightarrow k_{f} \mathrm{{\left( \mathrm{{[E]_T}} - \mathrm{{[ES]}}\right) [S]}} = k_{b}\mathrm{{[ES]}} + k_{\mathrm {cat}}\mathrm{{[ES]}} \nonumber \\&\Rightarrow k_{f}\mathrm{{[E]_T [S]}} = \left( k_{f}\mathrm{{[ES]}}\mathrm{{[S]}}+k_{b}\mathrm{{[ES]}} + k_{\mathrm {cat}}\mathrm{{[ES]}}\right) \nonumber \\&\Rightarrow \mathrm{{[ES]}} = \mathrm{{[E]_T}}\frac{\mathrm{{[S]}}}{\mathrm{{[S]}}+K_{\mathrm {M}}} \end{aligned}$$

(5.49)

where $K_{\mathrm {M}}=\left( k_{b}+k_{\mathrm {cat}}\right) /k_{f}$ is the Michaelis–Menten constant of the reaction which quantifies the affinity of the enzyme to bind to the substrate.

Regardless of the hypothesis made to find an expression for $\mathrm{{[ES]}}$, the rate $v_{P}$ of production of $\mathrm{{P}}$ can always be written as

$$\begin{aligned} v_{P} = \frac{\mathrm {d}\mathrm{{[P]}}}{\mathrm {d}t} = k_{\mathrm {cat}}\mathrm{{[ES]}} = k_{\mathrm {cat}}\mathrm{{[E]_T}}\frac{\mathrm{{[S]}}}{\mathrm{{[S]}}+K_{0.5}} = v_{max}\frac{\mathrm{{[S]}}}{\mathrm{{[S]}}+K_{0.5}} \end{aligned}$$

(5.50)

where $v_{max}=k_{\mathrm {cat}}\mathrm{{[E]_T}}$ is the maximal rate of production of $\mathrm{{P}}$ in the presence of enzyme saturation, when all the available enzyme takes part in the reaction; and the affinity constant $K_{0.5}$ equals the dissociation constant $K_{d}$ of the initial binding reaction in the chemical equilibrium approximation (Eq. 5.48), or the Michaelis–Menten constant in the QSSA (Eq. 5.49).

An important corollary of the Michaelis–Menten model of enzyme kinetics is that the fraction of the total enzyme that forms the intermediate complex $\mathrm{{ES}}$ can be expressed by a Hill equation of the type

$$\begin{aligned} \frac{\mathrm{{[ES]}}}{\mathrm{{[E]_T}}} = \frac{\mathrm{{[S]}}}{\mathrm{{[S]}}+K_{0.5}} = \mathcal {H}_{1}\left( [S],K_{0.5}\right) \end{aligned}$$

(5.51)

and $K_{0.5}$ can be regarded as the half-saturating substrate concentration of the reaction. Similarly, the effective reaction rate $v_{P}$ (Eq. 5.51) is proportional to the maximal reaction rate by a Hill-like term $\mathcal {H}_{1}\left( [S],K_{0.5}\right)$.

Appendix 2 Parameter Estimation

1.1 Appendix 2.1 Metabotropic Receptors

Rate constants $O_{N},\,\Omega _{N}$ (Eq. 5.14) lump information on astrocytic metabotropic receptors’ activation and inactivation, namely how long it takes for these receptors, once bound by the agonist, to trigger PLC$\upbeta$-mediated $\mathrm{{IP}_3}$ production and how long this latter lasts. Since $\mathrm{{IP}_3}$ production mediated by agonist binding with the receptors controls the initial intracellular $\mathrm{{Ca^{2+}}}$ surge, these two rate constants may be estimated by rise times of agonist-triggered $\mathrm{{Ca^{2+}}}$ signals. With this regard, experiments reported that application of 50$\, \upmu$ m DHPG—a potent agonist of mGluR5 which are the main type of metabotropic glutamate receptors expressed by astrocytes (Aronica et al. 2003)—triggers submembrane $\mathrm{{Ca^{2+}}}$ signals characterized by a rise time $\tau _{r}=0.272\pm {0.095}\,\mathrm{s}$. Because mGluR5 affinity ($K_{0.5}$) for DHPG is ${\sim }{2}\,{\, \upmu \textsc {m}}$ (Brabet et al. 1995), that is much smaller than the applied agonist concentration, receptor saturation may be assumed in those experiments whereby the receptor activation rate by DHPG ($O_{\text {DHPG}}$) can be expressed as a function of $\tau _{r}$ (Barbour 2001), i.e., $O_{\text {DHPG}}\approx \tau _{r}/({50}\,{\, \upmu \textsc {m}})$ = 0.055–0.113 ${\, \upmu \textsc {m}^{-1} \mathrm{s}^{-1}}$, so that $\Omega _{\text {DHPG}}=O_{\text {DHPG}}K_{0.5}\approx$ 0.11–0.22 ${\mathrm{s}^{-1}}$. Corresponding rate constants for glutamate may then be estimated assuming similar kinetics, yet with $K_{0.5}=K_N=\Omega _N/O_N \approx$ 3–10 ${\, \upmu \textsc {m}}$ (Daggett et al. 1995), that is 1.5–5-fold larger than $K_{0.5}$ for DHPG. Moreover, since rise times of $\mathrm{{Ca^{2+}}}$ signals triggered by nonsaturating physiological stimulation are faster than in the case of DHPG (Panatier et al. 2011), it may be assumed that $O_N > O_\mathrm {DHPG}$. With this regard, for a choice of $O_N \approx 3\times O_\text {DHPG}= {0.3}\,{\, \upmu \textsc {m}^{-1} \mathrm{s}^{-1}}$, with $K_{N}={6}{\, \upmu \textsc {m}}$ such that $\Omega _N = ({0.3}{\, \upmu \textsc {m}^{-1}\,\mathrm{s}^{-1}})({6}{\, \upmu \textsc {m}})={1.8}\,{\mathrm{s}^{-1}}$, a peak of extracellular glutamate concentration of ${250}\,{\, \upmu \textsc {m}}$, delivered at $t=0$ and exponentially decaying at rate $\Omega _{c}={40}\,{\mathrm{s}^{-1}}$ (Clements et al. 1992), is consistent with a peak fraction of bound receptors of ${\sim }0.75$ within ${\sim }{70}\,{\mathrm{m}\mathrm{s}}$ from stimulation (Eq. 5.14), which is in good agreement with experimental rise times.

1.2 Appendix 2.2 $\mathrm{{IP}_3}$R Kinetics

We consider a steady-state receptor open probability in the form of $p_\mathrm {open}(C,I) = \mathcal {H}_1^3(I,d_1)\cdot \mathcal {H}_1^3(C,d_5) (1-\mathcal {H}_{1}\left( C,Q_2\right) )^3$ with $Q_2=d_2(I+d_1)/(I+d_3)$ (see Chap. 3) and choose parameters to fit corresponding experimental data by Ramos-Franco et al. (2000) for (i) different $\mathrm{{Ca^{2+}}}$ concentrations $\hat{C}$ at a fixed $\mathrm{{IP}_3}$ level of $\bar{I}={1}{\, \upmu \textsc {m}}$, i.e., $\hat{p}(\hat{C})$; and (ii) for different $\mathrm{{IP}_3}$ concentrations ($\hat{I}$) at an intracellular $\mathrm{{Ca^{2+}}}$ concentration of $\bar{C}={25}\,\mathrm{{n} \textsc {m}}$, i.e., $\hat{p}(\hat{I})$. To reduce the problem dimensionality while retaining essential dynamical features of $\mathrm{{IP}_3}$ gating kinetics, we set $d_1=d_3$ (Li and Rinzel 1994). Accordingly, defining the vector parameter $\mathbf {x}_p = \left( d_1,d_2,d_5,O_2\right)$, we minimize the cost function $c_p(\mathbf {x}_p)=(p_\mathrm {open}(\hat{C},\bar{I})-\hat{p}(\hat{C}))^2 + (p_\mathrm {open}(\bar{C},\hat{I})-\hat{p}(\hat{I}))^2$ by the Artificial Bee Colony (ABC) algorithm (Karaboga and Basturk 2007) considering 2000 evolutions of a colony of 100 individuals.

Ultrastructural analysis of astrocytes in situ revealed that the probability of ER localization in the cytoplasmic space at the soma is between ${\sim }$40 and 70% (Pivneva et al 2008). This suggests that the corresponding ratio between ER and cytoplasmic volumes ($\rho _{A}$) is comprised between ${\sim }$0.4 and 0.7.

To estimate the cell’s total free $\mathrm{{Ca^{2+}}}$ content $C_{T}$ we make the consideration that the resting $\mathrm{{Ca^{2+}}}$ concentration in the cytosol is $<{0.15}\,{\, \upmu \textsc {m}}$ (Zheng et al. 2015) and can be neglected with respect to the amount of $\mathrm{{Ca^{2+}}}$ stored in the ER ($C_{ER}$) (Berridge et al. 2003). Hence, with $C_{ER}\ge {10}\,{\, \upmu \textsc {m}}$ (Golovina and Blaustein 1997) and a choice of $\rho _{A}\ge 0.4$, it follows that $C_{T}\approx \rho _{A}\, C_{ER} \ge {4}\,{\, \upmu \textsc {m}}$. In conditions close to store depletion during oscillations (Camello et al. 2002), this latter value would also coincide with the peak $\mathrm{{Ca^{2+}}}$ reached in the cytoplasm, which is reported between $5\,{\, \upmu \textsc {m}}$ and ${\sim }{20}\,{\, \upmu \textsc {m}}$ (Csordàs et al. 1999; Parpura and Haydon 2000; Kang and Othmer 2009; Shigetomi et al. 2010).

In our simulations, we set $\rho _A = 0.5$ while leaving arbitrary the choice of $C_T$ as far as the resulting $\mathrm{{Ca^{2+}}}$ oscillations qualitatively resemble the shape of those observed in experiments. The remaining parameters for CICR, i.e., $\mathbf {z}_c = \left( \Omega _C, O_P\right)$, were chosen to approximate the number and period of $\mathrm{{Ca^{2+}}}$ oscillations observed on average in experiments on cultured astrocytes that were stimulated by glutamate perfusion. By “on average,” we mean that we considered the average trace resulting from $n=5$ different $\mathrm{{Ca^{2+}}}$ signals generated within the same period of time and by the same stimulus in identical experimental conditions.

1.3 Appendix 2.3 $\mathrm{{IP}_3}$ signaling

Once set the CICR parameters, individual $\mathrm{{Ca^{2+}}}$ traces used to obtained the above-mentioned “average trace” were used to search for $\mathbf {z}_p = \left( O_\beta ,O_\delta ,O_{3K},\Omega _{5P}\right)$, assuming random initial conditions. The ensuing parameter values were also used in Figs. 5.4, 5.5, and 5.6 although $O_\beta ,\,O_\delta$, and $O_{3K}$ were increased, from case to case, by a factor comprised between 1.2 and 2 either to expand the oscillatory range or to promote CICR emergence (by increasing $O_\beta ,\,O_\delta$) or termination thereof (by larger $O_{3K}$ values).

1.4 Appendix 2.4 $\mathrm{{cPKC}}$ and DAG Signaling

Calcium-dependent cPKC-mediated phosphorylation has been documented for astrocytic mGluRs and $\mathrm{{P_2Y_1Rs}}$ (Codazzi et al. 2001; Hardy et al. 2005) and results in a reduction of receptor binding affinity by a factor $\zeta \approx$ 2–10 (Hardy et al. 2005), or possibly higher depending on the cell’s expression of cPKCs (Nakahara et al. 1997; Shinohara et al. 2011). Since experiments showed that cPKC is robustly activated only when $\mathrm{{Ca^{2+}}}$ increases beyond half of the peak concentration reached during oscillations (Codazzi et al. 2001) then, considering peak $\mathrm{{Ca^{2+}}}$ values of ${\sim }$ 1–3 ${\, \upmu \textsc {m}}$ (Shigetomi et al. 2010) allows estimating $\mathrm{{Ca^{2+}}}$ affinity of cPKC in the range of $K_{KC} \le 0.5--1.5\,{\, \upmu \textsc {m}}$ which indeed comprises the value of ${\sim }{700}\,{\mathrm{n}\textsc {m}}$ predicted experimentally (Mosior and Epand 1994). Of the same order of magnitude also is the $\mathrm{{Ca^{2+}}}$ affinity reported for DAGK, i.e., $K_{DC}\, {\approx }\,$0.3–0.4 ${\, \upmu \textsc {m}}$ (Sakane et al. 1991; Yamada et al. 1997).

Reported values of DAG affinities for cPKC and DAGK may considerably differ. Micellar assays of cPKCs activity suggests values of $K_{KD}$ as low as 4.6–13.3 nm (Ananthanarayanan et al. 2003), whereas studies on purified DAGK suggest a substrate affinity for this kinase of $K_{DD}\approx {60}\,{\, \upmu \textsc {m}}$ (Kanoh et al. 1983). The differences in experimental setups and the possibility that the activity of these kinases could be widely regulated by different DAG pools make these estimate of scarce utility for our model, where the DAG concentration is of the same order of magnitude of $\mathrm{{IP}_3}$ one. With this regard, we choose to set these affinities to 0.1 $\, \upmu$ m which corresponded in our simulations to the average intracellular DAG concentration during $\mathrm{{Ca^{2+}}}$ oscillations.

The remaining parameters, namely $\mathbf {z}_k = \left( O_{KD}, O_K, \Omega _D, O_D, \Omega _D\right)$, were arbitrarily chosen considering two constrains: (i) DAG concentration for damped $\mathrm{{Ca^{2+}}}$ oscillations must stabilize to a constant value; and (ii) the down phase of $\mathrm{{cPKC^*}}$ oscillations must follow that of $\mathrm{{Ca^{2+}}}$ ones as suggested by experimental observations by Codazzi et al. (2001).

Appendix 3 Software

The Python file used to generate the figures of this chapter can be downloaded from the online book repository at https://github.com/mdepitta/comp-glia-book/tree/master/Ch5.DePitta. The software for this chapter is organized in two folders. The folder contains data to fit the $G$-$ChI$ model. WebPlotDigitizer 4.0 (https://automeris.io/WebPlotDigitizer) was used to extract experimental data by Ramos-Franco et al. (2000, Figs. 6 and 7) and Codazzi et al. (2001, Fig. 5). The Jupyter notebook file found in this folder contains the code to load and clean experimental data used in the simulations.

The folder contains instead all the routines (including ) used for the simulations of this chapter. The two files and contains the core $G$-$ChI$ model implementation in C/C++11, while the class in provides the Python interface to simulate the $G$-$ChI$ model. The model was integrated by a variable-coefficient linear multistep Adams method in Nordsieck form which proved robust to correctly solve stiff problems rising from different parameter choices (Skeel 1986). Model fitting is provided by and relies on the PyGMO 2.6 optimization package (https://github.com/esa/pagmo2.git).

The library provides routines to estimate the period and range of $\mathrm{{Ca^{2+}}}$ oscillation as in Fig. 5.6. 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.