Wave failure at strong coupling in intracellular $\mathrm{C}{\mathrm{a}}^{2+}$ signaling system with clustered channels

As an important intracellular signal, $\mathrm{C}{\mathrm{a}}^{2+}$ ions control diverse cellular functions. In this paper, we discuss the $\mathrm{C}{\mathrm{a}}^{2+}$ signaling with a two-dimensional model in which the inositol 1,4,5-trisphosphate ($\mathrm{I}{\mathrm{P}}_3$) receptor channels are distributed in clusters on the endoplasmic reticulum membrane. The wave failure at large $\mathrm{C}{\mathrm{a}}^{2+}$ diffusion coupling is discussed in detail in the model. We show that with varying model parameters the wave failure is a robust behavior with either deterministic or stochastic channel dynamics. We suggest that the wave failure should be a general behavior in inhomogeneous diffusing systems with clustered excitable regions and may occur in biological $\mathrm{C}{\mathrm{a}}^{2+}$ signaling systems.

Figure 1

Bifurcation diagrams of two-dimensional $\mathrm{C}{\mathrm{a}}^{2+}$ diffusion model, i.e., the maximal and minimal concentrations of $\langle C\rangle$, against $p$ at $D=0.001\mu {\mathrm{m}}^2/\mathrm{s}$ (a) and at $D=200\mu {\mathrm{m}}^2/\mathrm{s}$ (b). The inset in (a) is an enlarged part for $p$ in the range of (0, $0.2\mu \mathrm{m}$).

Figure 2

Bifurcation diagrams of $C$ against $p$ for the point models with Eqs. (6) and (7) (a) and with Eqs. (6) and (8) (b). The inset in (a) is the enlarged part for $p$ in the range between 0 and $0.2\mu \mathrm{M}$. Solid lines are for stable states and dashed lines for unstable states.

Figure 3

Bifurcation diagrams of $\langle C\rangle$ against $p$ for the $\mathrm{C}{\mathrm{a}}^{2+}$ diffusion model at different $D$. In (a)–(d), $D=1$, 10, 20, and $50\mu {\mathrm{m}}^2/\mathrm{s}$, respectively.

Figure 4

The bifurcation diagrams of $\langle C\rangle$ against $p$ for the $\mathrm{C}{\mathrm{a}}^{2+}$ diffusion model at different $D$. In (a)–(d), $D=12$, 14, 14.6, and $15\mu {\mathrm{m}}^2/\mathrm{s}$, respectively.

Figure 5

The trajectories $\langle C\rangle$ of bistable oscillation states of the $\mathrm{C}{\mathrm{a}}^{2+}$ diffusion model at $D=15\mu {\mathrm{m}}^2/\mathrm{s}$ and $p=0.35\mu \mathrm{M}$.

Figure 6

(a)–(c) Bifurcation diagrams of $\langle C\rangle$ against $D$ for the $\mathrm{C}{\mathrm{a}}^{2+}$ diffusion model at $p=0.3$, 0.2, and 0.1 μM, respectively. The inset in (c) is the enlarged part for $D$ in the range of (0, $5\mu {\mathrm{m}}^2/\mathrm{s}$). (d) The oscillation frequency of $\mathrm{C}{\mathrm{a}}^{2+}$ wave against $D$ at $p=0.3$ (blue line), 0.2 (red line), and $0.1\mu \mathrm{M}$ (black line).

Figure 7

Robustness of wave failure with varying model parameters. Bifurcation diagrams of $\langle C\rangle$ against $D$ for the $\mathrm{C}{\mathrm{a}}^{2+}$ diffusion model at different maximum channel flux with $v_C=13.0$ (a) and $17.3{\mathrm{s}}^{-1}$ (b) and different maximum pump flux with $v_P=0.45$ (c) and $0.8{\mathrm{s}}^{-1}$ (d), respectively.

Figure 8

The snapshots of stochastic propagation wave at $p=0.5\mu \mathrm{M}$ and $D=15\mu {\mathrm{m}}^2/\mathrm{s}$. The time interval is 2 s between two successive snapshots.

Figure 9

Bifurcation diagrams of $\langle C\rangle$ against $p$ for the noisy $\mathrm{C}{\mathrm{a}}^{2+}$ diffusion model at different $D$. In (a)–(d), $D=10$, 15, 20, and $50\mu {\mathrm{m}}^2/\mathrm{s}$, respectively.

Figure 10

Bifurcation diagrams of $\langle C\rangle$ against $D$ for the noisy $\mathrm{C}{\mathrm{a}}^{2+}$ diffusion model at $p=0.3\mu \mathrm{M}$ and $N=36$ (a), $p=0.2\mu \mathrm{M}$ and $N=36$ (b), $p=0.25\mu \mathrm{M}$ and $N=25$ (c), and $p=0.45\mu \mathrm{M}$ and $N=15$ (d), respectively.