Population Density and Moment-based Approaches to Modeling Domain Calcium-mediated Inactivation of L-type Calcium Channels

Abstract

We present a population density and moment-based description of the stochastic dynamics of domain ${\text{Ca}}^{2+}$-mediated inactivation of L-type ${\text{Ca}}^{2+}$ channels. Our approach accounts for the effect of heterogeneity of local ${\text{Ca}}^{2+}$ signals on whole cell ${\text{Ca}}^{2+}$ currents; however, in contrast with prior work, e.g., Sherman et al. (Biophys J 58(4):985–995, 1990), we do not assume that ${\text{Ca}}^{2+}$ domain formation and collapse are fast compared to channel gating. We demonstrate the population density and moment-based modeling approaches using a 12-state Markov chain model of an L-type ${\text{Ca}}^{2+}$ channel introduced by Greenstein and Winslow (Biophys J 83(6):2918–2945, 2002). Simulated whole cell voltage clamp responses yield an inactivation function for the whole cell ${\text{Ca}}^{2+}$ current that agrees with the traditional approach when domain dynamics are fast. We analyze the voltage-dependence of ${\text{Ca}}^{2+}$ inactivation that may occur via slow heterogeneous domain [${\text{Ca}}^{2+}$]. Next, we find that when channel permeability is held constant, ${\text{Ca}}^{2+}$-mediated inactivation of L-type channels increases as the domain time constant increases, because a slow domain collapse rate leads to increased mean domain [${\text{Ca}}^{2+}$] near open channels; conversely, when the maximum domain [${\text{Ca}}^{2+}$] is held constant, inactivation decreases as the domain time constant increases. Comparison of simulation results using population densities and moment equations confirms the computational efficiency of the moment-based approach, and enables the validation of two distinct methods of truncating and closing the open system of moment equations. In general, a slow domain time constant requires higher order moment truncation for agreement between moment-based and population density simulations.