Abstract
We present a probability density approach to modeling localized Ca$^2+$
influx via L-type Ca$^2+$ channels and Ca$^2+$-induced Ca$^2+$
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 Ca$^2+$ conditioned
on "Ca$^2+$ 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 Ca$^2+$,
a realistic but minimal model of cardiac excitation-contraction coupling
is produced. Modeling Ca$^2+$ release unit activity using this
probability density approach avoids the computationally demanding
task of resolving spatial aspects of global Ca$^2+$ signaling,
while accurately representing heterogeneous local Ca$^2+$ 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 Ca$^2+$ 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 Ca$^2+$
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 Ca$^2+$ is in quasistatic equilibrium
with junctional sarcoplasmic reticulum Ca$^2+$ and, consequently,
univariate rather than multivariate probability densities may be
employed.
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