Inherent rhythm of smooth muscle cells in rat mesenteric arterioles: An eigensystem formulation

On the basis of experimental data and mathematical equations in the literature, we remodel the ionic dynamics of smooth muscle cells (SMCs) as an eigensystem formulation, which is valid for investigating finite variations of variables from the equilibrium such as in common experimental operations. This algorithm provides an alternate viewpoint from frequency-domain analysis and enables one to probe functionalities of SMCs' rhythm by means of a resonance-related mechanism. Numerical results show three types of calcium oscillations of SMCs in mesenteric arterioles: spontaneous calcium oscillation, agonist-dependent calcium oscillation, and agonist-dependent calcium spike. For simple single and double SMCs, we demonstrate properties of synchronization among complex signals related to calcium oscillations, and show different correlation relations between calcium and voltage signals for various synchronization and resonance conditions. For practical cell clusters, our analyses indicate that the rhythm of SMCs could (1) benefit enhancements of signal communications among remote cells, (2) respond to a significant calcium peaking against transient stimulations for triggering globally oscillating modes, and (3) characterize the globally oscillating modes via frog-leap (non-molecular-diffusion) calcium waves across inhomogeneous SMCs.

Figure 1

Schematic diagram of model components for mesenteric smooth muscle cell. ${\mathrm{BK}}_{\mathrm{Ca}}$: Large conductance calcium-activated ${\mathrm{K}}^{+}$ channel; ${\mathrm{K}}_{\mathrm{leak}}$: unspecified ${\mathrm{K}}^{+}$ leak channel; ${\mathrm{K}}_{\mathrm{v}}$: voltage-dependent ${\mathrm{K}}^{+}$ channel; ${\mathrm{Cl}}_{\mathrm{Ca}}$: calcium-activated ${\mathrm{Cl}}^{-}$ channel; $\mathrm{NSC}$: nonselective cation channel; $\mathrm{SOC}$: store-operated calcium-permeable nonselective cation channel; $\mathrm{VOCC}$: L-type voltage-operated ${\mathrm{Ca}}^{2+}$ channel; $\mathrm{NaK}$: ${\mathrm{Na}}^{+}\text{−}{\mathrm{K}}^{+}$-ATPase; $\mathrm{PMCA}$: plasma membrane ${\mathrm{Ca}}^{2+}$-ATPase; $\mathrm{NaKCl}$: ${\mathrm{Na}}^{+}\text{−}{\mathrm{K}}^{+}\text{−}{\mathrm{Cl}}^{-}$ cotransport; $\mathrm{NCX}$: ${\mathrm{Na}}^{+}\text{−}{\mathrm{Ca}}^{2+}$ exchange; $\mathrm{SR}$: sarcoplasmic reticulum; ${\mathrm{IP}}_3\mathrm{R}$: ${\text{IP}}_3$ receptor; $\mathrm{RyR}$: ryanodine receptor; $\mathrm{SERCA}$: SR ${\mathrm{Ca}}^{2+}$-ATPase pumps; $\mathrm{CSQN}$: calsequestrin; $\mathrm{CM}$: calmodulin; $\mathrm{R}$: ${\alpha }_1$-adrenoceptor; $\mathrm{G}$: G protein; $\mathrm{PLC}$: phospholipase C; $\mathrm{sGC}$: soluble guanylate cyclase; $\mathrm{GJ}$: nonselective gap-junction channel.

Figure 2

Numerical analysis of eigenvalues and eigenfunctions for a single SMC on the control condition: (a) real parts of eigenvalues, (b) imaginary parts of eigenvalues, (c) oscillating amplitude of transient variables in eigenmodes 22 and 23, and (d) oscillating phase of transient variables in eigenmodes 22 and 23.

Figure 3

Time evolutions of several transient variables in eigenmodes $(22,23)$ for a single SMC: (a) time evolution of ${\left[{\text{Ca}}^{2+}\right]}_i$ with given delta-function calcium stimulation at equilibrium $t=4.8\times {10}^4$ s, (b) transient variations of ${\left[{\text{Ca}}^{2+}\right]}_i$ after the delta-function calcium stimulation for the control condition ${\left[{\text{K}}^{+}\right]}_e=35.8$ mM and experimental condition ${\left[{\text{K}}^{+}\right]}_e=40.0$ mM, and (c) and (d) transient variations of several variables after the delta-function calcium stimulation.

Figure 4

Eigenvalue spectrum of a single SMC in response to extracellular potassium concentrations. (a) Real parts of eigenvalues for evolution time $\sim 1/{\omega }_{i,r}$, and (b) imaginary parts of eigenvalues for oscillation periods $\sim 2\pi /{\omega }_{i,c}$.

Figure 5

Oscillating amplitude ${\widehat{\delta }}_{\nu i}\equiv |{\delta }_{\nu i}/{\overline{\delta }}_{\nu i}|$ of variables in eigenmode $\left(22,23\right)$ in response to extracellular potassium concentrations.

Figure 6

Oscillating phase ${\vartheta }_{\nu i}$ of variables in eigenmode $\left(22,23\right)$ in response to extracellular potassium concentrations.

Figure 7

Oscillating amplitude of current $\mathrm{\Delta }{I}_i$ in eigenmode $\left(22,23\right)$ in response to extracellular potassium concentrations, in which ${\left[{\delta }_{\text{Ca}}\right]}_i=2\times {10}^{-5}$ mM.

Figure 8

Phase lag regarding oscillations for current $\mathrm{\Delta }{I}_i$ in eigenmode $\left(22,23\right)$ versus extracellular potassium concentrations, in which ${\left[{\delta }_{\text{Ca}}\right]}_i=2\times {10}^{-5}$ mM.

Figure 9

Schematic sketch of coupled time flows of signaling pathways for (a) spontaneous calcium oscillation and (b) agonist-dependent calcium oscillation. The curves show the timings of the maximum values of ionic concentrations ${\delta }_{\nu i\in \{1..6\}}$ and membrane potential ${\delta }_{\nu i\in \left\{7\right\}}$. The positions of the arrows schedule the time when the maximal amplitudes of current variations occur, and the lengths of the arrows indicate the oscillation amplitudes of channel currents. Arrows are colored as relevant ions. The opposite-direction current variations occur after a time lag $T_{\text{period}}/2=\pi /{\omega }_{22,i}$ and are not shown here. It is emphasized that the positive and negative arrows represent the relatively increasing and decreasing concentration to the values in equilibrium, respectively, and not the absolute values of concentrations.

Figure 10

Time-lag cross-correlation between calcium oscillation ${\left[{\delta }_{\text{Ca}}\right]}_i$ and current oscillations $\mathrm{\Delta }{I}_i$.

Figure 11

Time-lag cross-correlation between calcium oscillation ${\left[{\delta }_{\text{Ca}}\right]}_i$ and other transient variables ${\delta }_{\nu i}$.

Figure 12

Schematic sketch of coupled time flows of signaling pathways for eigenmode A, with $T_{\text{period}}=2\pi /{\omega }_{A,i}$. Relevant definitions for the curves and arrows are similar to those in Fig. 9.

Figure 13

Schematic sketch of coupled time flows of signaling pathways for eigenmode B, with $T_{\text{period}}=2\pi /{\omega }_{B,i}$. Relevant definitions for the curves and arrows are similar to those in Fig. 9.

Figure 14

Time evolutions of ${\left[\text{Ca}\right]}_i$ and $V_m$ for two coupled SMCs (I and II) at (a) on-resonance, (b) near-resonance, and (c) off-resonance conditions. Note that all voltage values are scaled and shifted to be comparable with calcium values.

Figure 15

Frequency spectrum of homogeneous SMC clusters at different cell numbers: (a) real parts of eigenvalues related to growing or decaying time, (b) imaginary parts of eigenvalues related to oscillation periods.

Figure 16

Calcium responses against delta-function calcium stimulation to SMC1 for homogeneous 6-SMCs: (a) at ${\left[\text{K}\right]}_e=34.6$ (oscillation activation; see region II in Fig. 4), and (b) at ${\left[\text{K}\right]}_e=20.0$ (oscillation deactivation; see region I in Fig. 4).

Figure 17

Frequency spectrum of inhomogeneous SMC clusters at different cell numbers: (a) real parts of eigenvalues related to growing or decaying time, (b) imaginary parts of eigenvalues related to oscillation periods.

Figure 18

Calcium responses against delta-function calcium stimulation to SMC1 for inhomogeneous 6-SMCs: (a) at ${\left[\text{K}\right]}_e=35.0$ (oscillation activation; see region II in Fig. 4), and (b) at ${\left[\text{K}\right]}_e=20.0$ (oscillation deactivation; see region I in Fig. 4).