Nonlinear onset of calcium wave propagation in cardiac cells

Spontaneous calcium (Ca) waves in cardiac myocytes are known to underlie a wide range of cardiac arrhythmias. However, it is not understood which physiological parameters determine the onset of waves. In this study, we explore the relationship between Ca signaling between ion channels and the nucleation of Ca waves. In particular, we apply a master equation approach to analyze the stochastic interaction between neighboring clusters of ryanodine receptor (RyR) channels. Using this analysis, we show that signaling between clusters can be described as a barrier hopping process with exponential sensitivity to system parameters. A consequence of this feature is that the probability that Ca release at a cluster induces release at a neighboring cluster exhibits a sigmoid dependence on the Ca content in the cell. This nonlinearity originates from the regulation of RyR opening due to more than one Ca ion binding site, in conjunction with Ca mediated cooperativity between RyR channels in clusters. We apply a spatially distributed stochastic model of Ca cycling to analyze the physiological consequences of this nonlinearity, and show that it explains the sharp onset of Ca wave nucleation in cardiac cells. Furthermore, we show that this sharp onset can serve as a mechanism for Ca alternans under physiologically relevant conditions. Thus our findings identify the nonlinear features of Ca signaling which potentially underlie the onset of Ca waves and Ca alternans in cardiac cells.

Figure 1

(a) Schematic illustration of the spatial architecture of Ca signaling in a cardiac ventricular cell. Signaling between channels occurs within dyadic junctions distributed in the 3D volume of the cell. (b) Illustration of two nearest neighbor signaling units (CRUs) showing the subcellular compartments. Here the superscript $n$ denotes the $n\mathrm{th}$ CRU in a 3D grid representing the cell. (c) Spatial architecture of the cell interior showing $Z$ planes.

Figure 2

(a) The local Ca concentration $c_p^i$ at junction $i$ in which a Ca spark occurs. In this simulation a spark is induced in that junction by raising the local concentration above the threshold for spark activation. Here the initial JSR load is $c_{\mathrm{jsr}}=1000\mu \mathrm{M}$, and the Ca concentration rises to a peak of roughly $c_p\sim 300\mu \mathrm{M}$. (b) The local concentration $c_p^j$ at a nearest neighbor junction $j$ on the same $Z$ plane. Here the local flux due to the RyR cluster is set to zero so that the rise in concentration is due only to the diffusive flux from junction $i$. At this site the local concentration rises to $\sim 3\mu \mathrm{M}$.

Figure 3

(a) The effective potential $\mathrm{\Phi }\left(x\right)$ for spark off (black solid line, $s_{\mathrm{off}}=0.04$), and spark on (red dashed line, $s_{\mathrm{on}}=0.5$). As $s$ is increased the barrier height between the stable stationary point at $x_o$ and the unstable point at $x_1$ is reduced. (b) Plot of the stationary points $x_o$ and $x_1$ as a function of the parameter $s$.

Figure 4

(a) The effective potential barrier $N\mathrm{\Delta }\mathrm{\Phi }$ as a function of the JSR load $c_{\mathrm{jsr}}$ for the spark on case $s=0.5$. (b) The waiting time vs the JSR load computed using an exact stochastic simulation of an isolated cluster of $N=100$ RyR channels. (c) The transmission probability $p_{ij}$ as a function of the JSR load computed using the exact stochastic simulation (red dashed line) and from Eq. (10) (black solid line). (d) The threshold for transmission probability $c_{th}$ is computed as the JSR load where $p_{ij}\left(c_{th}\right)=1/2$, where $p_{ij}$ is computed with the exact stochastic simulation (black circles). $c_{\mathrm{jsr}}^{*}$ (red squares) is computed using our analytic approximation given by Eq. (13).

Figure 5

Ca wave nucleation and propagation in a 3D stochastic cell model. (a) Spatial distribution of dyadic junction Ca concentration $c_p^n$ visualized across a 2D cross section of a cell with $20\times 20\times 60$ CRUs. (b) The mean waiting time $T_w$ for a Ca wave as a function of JSR load. Points shown are averaged over 100 independent simulations. (c) The probability $p_w$ of a Ca wave occurring within a $200\mathrm{ms}$ time interval as a function of the JSR load (blue solid line). Line is computed by averaging over 100 independent simulations. The transmission probability between two adjacent units, denoted as $p_{ij}^r$, computed using the Restrepo model (red dashed line). The transmission probability computed using Eq. (10)) (black dash-dotted line). (d) The threshold as a function of the number of channels in the cluster $N$. Blue dash-dotted line corresponds to the wave propagation onset $c_w$ defined as $p_w\left(c_w\right)=1/2$. Red dashed line is the SR load such that $p_{ij}^r\left(\tilde{c}\right)=1/2$, computed directly from the Restrepo model, and the black solid line is the analytic threshold given by Eq. (13)).

Figure 6

(a) The calculated peak of diastolic Ca concentration, $c_i^{\max }$, as a function of $c_{\mathrm{jsr}}$ for cases where $30\%$ (black, bottom line), $50\%$ (red, middle line), and $70\%$ (top, blue line), of the clusters have LCC channels. (b) Time dependence of the steady-state JSR load simulated using a 3D cell model where only 30% of the clusters are driven by LCC channels. In this case the cell is paced with an AP clamp with a period of $T=200\mathrm{ms}$. The shape of the AP clamp is the same as that used in the original Restrepo model. (c) Simulated line scan of subcellular Ca release during alternans.

Figure 7

Plot of the peak Ca transient $c_i^{\max }$ for the last two beats after pacing the cell for 20 beats. Black circles and red squares correspond to 30% and 50% LCC density, respectively.