Physics of Spontaneous Calcium Oscillations in Cardiac Cells and Their Entrainment

Mechanical contraction in muscle cells requires Ca to allow myosin binding to actin. Beating cardiomyocytes contain internal Ca stores whose cytoplasmic concentration oscillates. Our theory explains observed single channel dynamics as well as cellular oscillations in spontaneously beating cardiomyocytes. The Ca dependence of channel activity responsible for Ca release includes positive feedback with a delayed response. We use this to predict a dynamical equation for global calcium oscillations with only a few physically relevant parameters. The theory accounts for the observed entrainment of beating to an oscillatory electric or mechanical field.

Figure 1

Schematic representation of calcium dynamics in cardiac cells. A ${\mathrm{Ca}}^{2+}$ influx from the environment to the cytoplasm occurs due to fluctuations in calcium channels and pump activity. This causes a release of stored ${\mathrm{Ca}}^{2+}$ from the sarcoplasmic reticulum (SR) through SR embedded RyR channels. The increase in cytoplasmic ${\mathrm{Ca}}^{2+}$ concentration facilitates both contraction of actomyosin in sarcomeres and further opening of RyR channels (green arrows). Eventually, RyR channels close, and various ion pumps and channels restore ${\mathrm{Ca}}^{2+}$ to its original concentration (dashed lines), and the cycle begins again.

Figure 2

Analytical and numerical calculations, and experimental data, showing the onset of synchronization as a function of amplitude ($a_p/a_c$) and frequency (${\omega }_p={\mathrm{\Omega }}_p/{\mathrm{\Omega }}_c$) ratios. Parameters were chosen at the edge of the physical range to explore the most dramatic changes. The black line is the analytical approximation of Eq. (12) (valid when ${\omega }_p\approx 1$). The blue dashed line connects the averages of the numerically calculated values (a guide to the eye). Below the curve (shaded area), the beating dynamics contains both the cell and the probe frequencies, while above the curve, cells are entrained to the external force (i.e., they beat with frequency ${\mathrm{\Omega }}_p$). Color legend: (blue) $\epsilon =-0.1,\mathrm{\Gamma }=5$, (orange) $\epsilon =-0.5,\mathrm{\Gamma }=1.4$, (green) $\epsilon =-0.02,\mathrm{\Gamma }=10$. Purple circles show the reproduced data from Ref. [34], showing entrainment for a fixed amplitude ratio estimated by the transition from entrained to nonentrained beating at ${\omega }_p\approx 2.5$. Inset: Same numerical results plotted for the amplitudes ratio $a_p/a(a_p)$, where $a(a_p)$ is evaluated directly from the numerical solution. Black curve given by Eq. (12) where the amplitude of oscillations $a$ is not approximated by $a_c$, but rather extracted from the numerical calculations.