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A Probability Density Approach to Modeling Local Control of Calcium-Induced Calcium Release in Cardiac Myocytes

George S. B. Williams ^*, Marco A. Huertas ^*, Eric A. Sobie  , M. Saleet Jafri   and Gregory D. Smith ^*

^* Department of Applied Science, College of William and Mary, Williamsburg, Virginia; Department of Bioinformatics and Computational Biology, George Mason University, Manassas, Virginia; Medical Biotechnology Center and the Institute of Molecular Cardiology, University of Maryland Biotechnology Institute, Baltimore, Maryland; and Department of Pediatrics, New York University School of Medicine, New York, New York

Correspondence: Address reprint requests to Gregory D. Smith, Dept. of Applied Science, McGlothlin-Street Hall, Rm. 305, College of William and Mary, Williamsburg, VA 23187. E-mail: greg{at}as.wm.edu.

We present a probability density approach to modeling localized Ca²⁺ influx via L-type Ca²⁺ channels and Ca²⁺-induced Ca²⁺ release mediated by clusters of ryanodine receptors during excitation-contraction coupling in cardiac myocytes. Coupled advection-reaction equations are derived relating the time-dependent probability density of subsarcolemmal subspace and junctional sarcoplasmic reticulum [Ca²⁺] conditioned on "Ca²⁺ release unit" state. When these equations are solved numerically using a high-resolution finite difference scheme and the resulting probability densities are coupled to ordinary differential equations for the bulk myoplasmic and sarcoplasmic reticulum [Ca²⁺], a realistic but minimal model of cardiac excitation-contraction coupling is produced. Modeling Ca²⁺ release unit activity using this probability density approach avoids the computationally demanding task of resolving spatial aspects of global Ca²⁺ signaling, while accurately representing heterogeneous local Ca²⁺ signals in a population of diadic subspaces and junctional sarcoplasmic reticulum depletion domains. The probability density approach is validated for a physiologically realistic number of Ca²⁺ release units and benchmarked for computational efficiency by comparison to traditional Monte Carlo simulations. In simulated voltage-clamp protocols, both the probability density and Monte Carlo approaches to modeling local control of excitation-contraction coupling produce high-gain Ca²⁺ release that is graded with changes in membrane potential, a phenomenon not exhibited by so-called "common pool" models. However, a probability density calculation can be significantly faster than the corresponding Monte Carlo simulation, especially when cellular parameters are such that diadic subspace [Ca²⁺] is in quasistatic equilibrium with junctional sarcoplasmic reticulum [Ca²⁺] and, consequently, univariate rather than multivariate probability densities may be employed.

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