Synergy Between Intercellular Communication and Intracellular Ca²⁺ Handling in Arrhythmogenesis

Abstract

Calcium is the primary signalling component of excitation-contraction coupling, the process linking electrical excitability of cardiac muscle cells to coordinated contraction of the heart. Understanding ${\text{Ca}}^{2+}$ handling processes at the cellular level and the role of intercellular communication in the emergence of multicellular synchronization are key aspects in the study of arrhythmias. To probe these mechanisms, we have simulated cellular interactions on large scale arrays that mimic cardiac tissue, and where individual cells are represented by a mathematical model of intracellular ${\text{Ca}}^{2+}$ dynamics. Theoretical predictions successfully reproduced experimental findings and provide novel insights on the action of two pharmacological agents (ionomycin and verapamil) that modulate ${\text{Ca}}^{2+}$ signalling pathways via distinct mechanisms. Computational results have demonstrated how transitions between local synchronisation events and large scale wave formation are affected by these agents. Entrainment phenomena are shown to be linked to both intracellular ${\text{Ca}}^{2+}$ and coupling-specific dynamics in a synergistic manner. The intrinsic variability of the cellular matrix is also shown to affect emergent patterns of rhythmicity, providing insights into the origins of arrhythmogenic ${\text{Ca}}^{2+}$ perturbations in cardiac tissue in situ.

Appendix

Mathematical Formulation

For each individual cell, the system is described by three variables: $x$ = $[{\text{Ca}}^{2+}]_i$ represent the cytosolic free ${\text{Ca}}^{2+}$ concentration, $y$ = $[{\text{Ca}}^{2+}]_{SR}$ the ${\text{Ca}}^{2+}$ concentration in the SR and $z$ the cell membrane potential.

$$\begin{aligned} \frac{dx}{dt}=\mathop {A}_{\begin{array}{c} \text {NSCC} \\ \text {influx} \end{array}}-\mathop {E_{\text{Ca}}\frac{z-z_{Ca1}}{1+e^{-(z-z_{Ca2})/R_{\text{Ca}}}}}_{\text {VOCC influx}}+\mathop {E_{Na/Ca}\frac{x}{x+x_{Na/Ca}}\left( z-z_{Na/Ca}\right) }_{\text {NCX}}\nonumber \\ -\mathop {B\frac{x^n}{x^n+x_b^n}}_{\text {SR uptake}}+\mathop {C_r\frac{x^{p_r}}{x^{p_r}+x_r^{p_r}}\frac{y^{m_r}}{y^{m_r}+y_r^{m_r}}}_{\text {RyR CICR}}-\mathop {Dx^k\left( 1+\frac{z-z_d}{R_d}\right) }_{{\text{Ca}}^{2+}\text {extrusion}}+\mathop {Ly}_{\text {SR leak}} \end{aligned}$$

(A-1a)

$$\begin{aligned} \frac{dy}{dt}=\mathop {B\frac{x^n}{x^n+x_b^n}}_{\text {SR uptake}}-\mathop {C_r\frac{x^{p_r}}{x^{p_r}+x_r^{p_r}}\frac{y^{m_r}}{y^{m_r}+y_r^{m_r}}}_{\text {RyR CICR}} \end{aligned}$$

(A-1b)

$$\begin{aligned} \frac{dz}{dt}=-\gamma \ \bigg (\mathop {E_{Cl}\frac{x}{x+x_{Cl}}\left( z-z_{Cl}\right) }_{{\text{Cl}}^{-}\text {channels}}+\mathop {2E_{\text{Ca}}\frac{z-z_{Ca1}}{1+e^{-(z-z_{Ca2})/R_{\text{Ca}}}}}_{\text {VOCC influx}}\nonumber \\ +\mathop {E_{Na/Ca}\frac{x}{x+x_{Na/Ca}}\left( z-z_{Na/Ca}\right) }_{\text {NCx}}+\mathop {E_K\left( z-z_{K}\right) \frac{x}{x+\beta e^{-(z-z_{Ca3})/R_K}}}_{{\text{K}}^{+}\text {efflux}}\bigg ) \end{aligned}$$

(A-1c)

The electric reversal potentials with respect to ${\text{Ca}}^{2+}$ and ${\text{Na}}^{+}$ are determined from the Nernst equation, see Parthimos et al. for details.29,30 The different terms and associated fixed parameter values can be found in Table 1. The subscript $r$ refers to RyR-mediated CICR, as opposed to InsP$_3$-induced ${\text{Ca}}^{2+}$ release (not included in the present formulation).

For each SMC $i$, the set of its nearest neighbours $j \in N_i$ consist of at most 6 neighbours, disposed at the vertices of an hexagon, depending on its position within the domain or at the boundary. Two terms:

$$\begin{aligned} J_{Ca,i}&=g_{\text{Ca}} \sum _{j \in N_i} \left( x_j-x_i\right) \qquad V_{m,i}&=g_z \sum _{j \in N_i} \left( z_j-z_i\right) \end{aligned}$$

(A-2)

are added to (A-1a) and (A-1c), respectively, to model ${\text{Ca}}^{2+}$ and electrical coupling.