Table 1.
Parameter values used in the simulations.
Here, refers to the extracellular domain,
refers to the membrane domain, and
refers to the intracellular domain (see Fig 1). Note that the intracellular diffusion coefficients for Na+, K+ and
are set up such that the ratio between the intracellular and extracellular diffusion coefficients are the same as for Ca2 + . Moreover, in the cell membrane (
), the diffusion coefficient is set to zero for all ions. Electrodiffusion through channels and exchangers in the membrane is handled using local fluxes as explained in Sect 2.4.
Fig 1.
Illustration of the considered domain.
A: In the cardiomyocyte, small domains called dyads are located in areas where the membrane of the SR is close to T-tubule membrane. B: In the dyad, Ca2 + channels and sodium-calcium exchangers (NCXs) in the cell membrane are directly apposed to ryadodine receptor channels (RyRs) in the membrane of the SR. C: The computational domain consists of an extracellular space, , a cell membrane,
, and an intracellular domain,
. A part of this domain is used to represent the dyad with an associated part of the cell membrane and extracellular space (T-tubule). D: In the computational domain, ion channels and exchangers occupy specific locations in the cell membrane.
Table 2.
Parameters characterizing the intracellular Ca2+ binding buffers.
The values are based on [31].
Table 3.
Default geometry parameter values used in the simulations.
For definitions of the lengths, see Fig 1C. Note that Li denotes the length from the membrane of the T-tubule to the membrane of the SR (dyad width), and will be used as a control parameter in computational experiments below.
Table 4.
Initial conditions for the ionic concentrations in the intracellular and extracellular domains.
Fig 2.
Illustration of two alternative representation of ion channels embedded in the membrane.
A: Ion channels are represented as subdomains of the membrane which selectively allow for electrodiffusion of certain ion species. For example, a K+ channel only allows for electrodiffusion of K+ ions. B: The flux through ion channels are represented as internal boundary conditions for the ion concentrations associated with the channel. In this representation, electrodiffusion in the channel is formulated as a boundary condition with a specified flux. This flux can be defined by the integration of the 1D Nernst equation for the specific ion specie under consideration.
Table 5.
Parameters for the channel and exchanger fluxes.
The single channel conductances, ,
, and
are taken from [35]. The NCX density,
, is taken from [36] and the remaining parameters characterize the NCX and are taken from [37].
Fig 3.
Illustration of the mesh applied in the simulations.
We show a slice of the mesh in the x- and y-directions. The mesh is refined near the membrane and near membrane channels and exchangers. Similar refinements are applied in the z-direction. The Ca2+ channel and the Na+/Ca2+-exchanger (NCX) are located in a volume referred to as the dyad. A K+ and a Na+ channel are located in another volume of interest in the simulations.
Fig 4.
Decay following a perturbation of the K+ concentration in a PNP simulation.
We consider a simple 2D example with two ions, K+ and . A: Initial conditions for the two ions. B: Concentrations in the center point as functions of time. C: Charge density,
, in the center point as a function of time. D: The equation terms,
and
, of (2) in the center point as functions of time during the first 4 ns of simulation (left) and after 4 ns (right). We use
,
ns and a uniform mesh with
nm.
Fig 5.
Formation of a Debye layer near a membrane in a PNP simulation.
We consider a simple 2D example with two ions. Initially, the concentrations are constant in and
. The initial concentration of
is 100 mM in both
and
, and the initial concentration of K+ is 100.1 mM in
and 99.9 mM in
. A: Ionic concentrations and charge density near the membrane as functions of x for five different points in time. B: Charge density,
, 0.1 nm right of the membrane (x1) and 20 nm to the right of the membrane (x2) as functions of time. C: Illustration of the simulation setup. We use
,
ns and a uniform mesh with
nm.
Fig 6.
Example illustration of the setup used to visualize the simulation results.
The setup is described in detail in Sect 3.3, and re-used in many figures below.
Fig 7.
Dynamics following the opening a K+ channel in a PNP model simulation.
The figure setup is described in Sect 3.3. We use s and an adaptive mesh like illustrated in Fig 3.
Fig 8.
Steady state solutions close to the membrane for the cell at rest.
Only a K+ channel is open. The plots show the solutions along a line in the x-direction, taken from the simulation shown in Fig 7.
Fig 9.
Dynamics following the opening of a Na+ channel in a PNP model simulation.
The figure setup is described in Sect 3.3. We use s and an adaptive mesh like illustrated in Fig 3.
Fig 10.
Dyad dynamics following the opening of a Ca2+ channel in a PNP model simulation.
The figure setup is described in Sect 3.3. We use s and an adaptive mesh like illustrated in Fig 3.
Fig 11.
Dyad dynamics following the opening of a Ca2+ channel in a PNP model simulation including an open NCX.
The figure setup is described in Sect 3.3. We use s and an adaptive mesh like illustrated in Fig 3.
Fig 12.
Dyad dynamics following the opening of an Na+ channel in a PNP model simulation including an NCX.
The Ca2+ channel is not opened. The figure setup is described in Sect 3.3. We use s and an adaptive mesh like illustrated in Fig 3.
Fig 13.
Dyad dynamics following the opening of a Na+ channel near an NCX in a PNP model simulation.
The Ca2+ channel is not opened. The figure setup is described in Sect 3.3. We use s and an adaptive mesh like illustrated in Fig 3.
Fig 14.
RyR activation time for different values of the dyad width (Li) and the intracellular Ca2+ diffusion coefficient ().
The RyR activation time is defined as time from the membrane Ca2+ channel is opened until the Ca2+ concentration outside of an apposing RyR channel reaches 0.5 M. We perform PNP model simulations similar to the one displayed in Fig 11. We use
ns and an adaptive mesh like illustrated in Fig 3.
Fig 15.
Decay following a perturbation of one of the concentrations in simulations of the PNP model and the pure diffusion model.
We consider the simple 2D example with two ions displayed in Fig 4. The initial conditions of the two ions are shown in Fig 4A, and we plot the concentrations in the point in the center of the domain, like in Fig 4B. We have used ns and a uniform mesh with
nm.
Fig 16.
Dyad dynamics following the opening of a Ca2+ channel in a pure reaction-diffusion model version of the PNP simulation displayed inFig 11.
The figure setup is described in Sect 3.3. We have used s and an adaptive mesh like illustrated in Fig 3.
Fig 17.
RyR activation time for different values of the dyad width (Li) and the intracellular Ca2+ diffusion coefficient () in the PNP and reaction-diffusion models.
The RyR activation time is defined as time from the membrane Ca2+ channel is opened until the Ca2+ concentration outside of an apposing RyR channel reaches 0.5 M. The PNP results are also displayed in Fig 14. We have used
ns and an adaptive mesh like illustrated in Fig 3. Note that the electrical potential computed in the PNP model is used to compute the transmembrane fluxes in the reaction-diffusion model.
Fig 18.
Terms in the PNP model equation (6) governing the Ca2+ concentration. We consider the solution in the first 80 ns of simulation after the Ca2+ channel is opened in points at different distances, d, in the x-direction from the intracellular mouth of the Ca2+ channel. Note that the scaling of the y-axis is different in each panel. We have used the default model parameters, ns, and an adaptive mesh like illustrated inFig 3.
Fig 19.
Illustration of two ODE model representations of the dyad dynamics.
The ionic concentrations are assumed to be constant (in space) in each considered compartment. A: The ODE model consists of one dyad compartment, , and one larger compartment representing the surrounding cytosol,
. B: The ODE model consists of two dyad compartments (one near the Ca2+ channel,
, and one near the RyR,
), in addition to the larger compartment representing the surrounding cytosol,
.
Fig 20.
Time from the Ca2+ channel is opened until the [Ca2+] outside of an apposing RyR channel reaches 0.5 M in simulations of the two ODE models illustrated in Fig 19 and in the PNP model (left).
We have used s. Note that in the leftmost panel, the results for different values of
overlap.