Acta Biotheoretica

, Volume 64, Issue 1, pp 11–32 | Cite as

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

  • Xiao Wang
  • Kiah Hardcastle
  • Seth H. Weinberg
  • Gregory D. Smith
Regular Article


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.


L-type \({\text{Ca}}^{2+}\) channel Population density model Moment-based model \({\text{Ca}}^{2+}\)-dependent inactivation 



The work was supported in part by National Science Foundation Grant DMS 1121606 to GDS and the Biomathematics Initiative at The College of William & Mary.


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Copyright information

© Springer Science+Business Media Dordrecht 2015

Authors and Affiliations

  • Xiao Wang
    • 1
  • Kiah Hardcastle
    • 1
  • Seth H. Weinberg
    • 1
  • Gregory D. Smith
    • 1
  1. 1.Department of Applied ScienceThe College of William & MaryWilliamsburgUSA

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