Figure 1.
Sketch of the signaling pathways considered in the ChI model.
(a) Ca2+-induced Ca2+ release (CICR) from the endoplasmic reticulum (ER) is the main mechanism responsible for intracellular Ca2+ dynamics in astrocytes. (b) Schematic of the coupling between Ca2+ dynamics and IP3 metabolism in the astrocyte. (c) Endogenous IP3 production is brought forth by hydrolysis of PIP2 by PLCδ (the activity of which is regulated by Ca2+). (d) Degradation of IP3 mainly occurs through IP3 3-kinase- (3K-) catalyzed phosphorylation and inositol polyphosphate 5-phosphatase (IP-5P)-mediated dephosphorylation. For simplicity, Ca2+-dependent PKC-mediated phosphorylation of IP3-3K [75] and competitive binding of IP4 to IP-5P are not considered in this study. The legend of different arrows is given below (c).
Figure 2.
The coupling functions used in the current study to model different types of gap-junctions.
Shown is the relative flux, i.e. the value the IP3 flux divided by the coupling force F as a function of the IP3 gradient (ΔIP3) between two coupled cells, for linear, threshold-linear and sigmoidal coupling. Parameters: IP3thr = 0.3 µM, IP3scale = 0.05 µM.
Figure 3.
Propagation patterns with linear (a, b) and non-linear sigmoidal (c, d) gap junctions.
The astrocyte chain was composed of 12 FM-encoding cells with reflective boundary conditions. Stimulation triggered by IP3bias = 1.0 µM from t = 0 s to t = 120 s applied to the first cell in the line.
Figure 4.
Traveled distance for the propagation of Ca2+ waves as a function of stimulus intensity.
The stimulus is applied to the first cell and the traveled distance is expressed in number of cells. With moderate coupling strength (F = 2 µm/s) and N = 25 cells, long-range Ca2+ propagation is observed in the case of FM (a) but not AFM chains (b). Linear gap junctions (closed green circles) do not sustain propagation over long distances, whatever the encoding mode is. Long-range propagation is observed for AFM cells coupled by non-linear sigmoid (open magenta circles) or threshold-linear (times signs) gap junctions. Variation of the propagation range with F for 50 coupled FM cells (IP3bias = 2.0 µM) is shown in (c) for sigmoid and linear gap junctions. Panel (d) shows a map of the r5P-νδ parameter space, where black dots correspond to waves propagating over all the cells, while white areas locate non-propagating waves. In these figures, a wave was considered to have reached a given cell whenever the amplitude of calcium variations in this cell was larger than 0.6 µM. Boundary conditions in these simulations were reflective.
Figure 5.
Propagation patterns for linear (a, b) and non-linear sigmoidal (c, d) gap junctions.
The astrocyte chain was composed of 12 AFM astrocytes. Stimulus protocol and other parameters as in Figure 3.
Figure 6.
Calcium (a–c) and IP3 (d–f) traces for wave propagation in composite astrocyte chains.
Black traces locate FM cells while AFM cells are displayed with gray traces. (a, d) Alternating one FM with one AFM cell for example, may not allow propagation beyond the second AFM cell. (b, e) Larger propagation distances can be obtained by increasing the maximal rate of PLCδ in AFM cells. (c, f) Alternatively, longer traveling distances can be observed in chains where two AFM cells are separated by more than one FM cell. Parameters: a stimulus of IP3bias = 1.2 µM was applied between t = 10 and t = 300 s on the first cell of the chain (i.e. cell 1). Other parameters: vδ(AFM) = 0.108 µM/s; vδ(FM) = 0.832 µM/s; r5P = 0.202 s−1; IP3thr = 0.215 µM; (b, e): vδ(AFM) = 0.15 µM/s. In all these simulations, a 5-minute-long stimulus was applied.
Figure 7.
Complex behaviors at low coupling strength (F = 0.23 µM·s−1).
The stimulus is an oscillatory input (positive square wave) applied to the central cell of an N = 41 cell chain. (a) Calcium concentration in cells 1 to 21 (cells 21 to 41, respectively, are identical). (b) The trajectory in the C-h-IP3 phase space for cell 21 (i.e. the stimulated cell) and (c, d) the corresponding Ca2+ and IP3 time series. Simulation performed on FM-encoding astrocytes with reflective boundary conditions and sigmoid gap junctions. Stimulus protocol: positive square wave of 50-second period and duty cycle of 0.4.