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Edited by: Johan Elon Hake, Simula Research Laboratory, Norway

Reviewed by: George S. B. Williams, University of Maryland, Baltimore, USA; Daisuke Sato, University of California, Davis, USA; Derek Rowland Laver, University of Newcastle, Australia

*Correspondence: Martin Falcke, Mathematical Cell Physiology, Max Delbrück Center for Molecular Medicine, Robert-Rössle-Straße 10, 13125 Berlin, Germany

This article was submitted to Biophysics, a section of the journal Frontiers in Physiology

This is an open-access article distributed under the terms of the Creative Commons Attribution License (CC BY). The use, distribution or reproduction in other forums is permitted, provided the original author(s) or licensor are credited and that the original publication in this journal is cited, in accordance with accepted academic practice. No use, distribution or reproduction is permitted which does not comply with these terms.

Mathematical modeling of excitation-contraction coupling (ECC) in ventricular cardiac myocytes is a multiscale problem, and it is therefore difficult to develop spatially detailed simulation tools. ECC involves gradients on the length scale of 100 nm in dyadic spaces and concentration profiles along the 100 μm of the whole cell, as well as the sub-millisecond time scale of local concentration changes and the change of lumenal Ca^{2+} content within tens of seconds. Our concept for a multiscale mathematical model of Ca^{2+} -induced Ca^{2+} release (CICR) and whole cardiomyocyte electrophysiology incorporates stochastic simulation of individual LC- and RyR-channels, spatially detailed concentration dynamics in dyadic clefts, rabbit membrane potential dynamics, and a system of partial differential equations for myoplasmic and lumenal free Ca^{2+} and Ca^{2+}-binding molecules in the bulk of the cell. We developed a novel computational approach to resolve the concentration gradients from dyadic space to cell level by using a quasistatic approximation within the dyad and finite element methods for integrating the partial differential equations. We show whole cell Ca^{2+}-concentration profiles using three previously published RyR-channel Markov schemes.

Cardiomyocyte muscle filament shortening and lengthening is a Ca^{2+} dependent process. The timing of contraction is controlled through electrical excitation via a process known as excitation-contraction-coupling (ECC). ECC is mediated through Ca^{2+}, and is facilitated through an amplification process known as Ca^{2+}-induced Ca^{2+} release (CICR). In ventricular myocytes, CICR is controlled locally by the colocalization of L-type Ca^{2+}-channels (LCCs) in the T-tubule membrane on the one side of a dyadic cleft [aka Ca^{2+} release unit (CRU)] and ryanodine receptor channels (RyRs) in the junctional sarcoplasmic reticulum (jSR) membrane on the other side. Depolarization of the plasma membrane leads to the activation of LCCs, which causes Ca^{2+} entry from the extracellular space into the dyadic space. The influx of Ca^{2+} activates RyRs, which release Ca^{2+} from the sarcoplasmic reticulum (SR) (Fabiato and Fabiato, ^{2+} -release-units, Ca^{2+} dynamics are distinct from the myoplasm, with steep [Ca^{2+}] gradients and several-fold higher [Ca^{2+}] (Stern, ^{2+} transient. The CRUs are coupled through myoplasmic Ca^{2+} diffusion, through SR Ca^{2+} diffusion, and through the spatially homogeneous membrane voltage.

Fundamental properties of ECC/CICR are: (i) that through a mechanism of local control of CICR, a graded LCC current produces a graded RyR Ca^{2+} release (Barcenas-Ruiz and Wier, ^{2+} release in highly localized cell-subcompartments, with dyadic space [Ca^{2+}] rising to ~1000 × myoplasmic [Ca^{2+}] (Koh et al.,

These properties illustrate that multiple length scales (tens of nanometers in the dyadic space to 100 μm cell size) and time scales (sub-millisecond for [Ca^{2+}] changes in the dyad to tens of seconds for SR dynamics) are involved. To account for that multi-scale character, recently several models with spatially distributed Ca^{2+} release sites have been developed (Restrepo et al., ^{2+} dynamics and at gaining independence from model-simplifying assumptions as far as possible with reasonable effort. Many of these models represent the dyadic space by a single compartment not resolving concentration gradients. Detailed spatially resolved models of the CRU have been developed which represent the steep local [Ca^{2+}] gradients in the cleft (Koh et al.,

We use mathematical multiscale techniques (Green Function, quasistatic approximation) to simulate a computationally efficient mathematical model of CRUs with spatially resolved [Ca^{2+}] and stochastic state dynamics of all individual LC- and RyR-channels. The dynamics of up to 5120 CRUs is then embedded into simulations of the cellular concentration fields for Ca^{2+} and Ca^{2+}-binding molecules as well as membrane potential time course.

The mathematical model comprises a system of partial differential equations for the cytosolic and sarcoplasmic concentration dynamics, _{c} models for the individual CRUs and a system of ordinary differential equations for the electrophysiology (see Figure

^{2+}, cytosolic and sarcoplasmic mobile buffers and a cytosolic stationary buffer. The _{c} Ca^{2+} release units (CRUs) are simulated all individually and are source terms in the bulk concentration dynamics PDEs. The state dynamics of each of their LC- or RyR-channels is a continuous time Markov chain. The concentration profile in the dyadic space is modeled in spatial detail with a quasistationary approximation, the dynamics of the concentrations of free Ca^{2+} and buffer in the jSR are determined by release into the cleft and refilling from the network SR (nSR). The electrophysiology model has been developed by Mahajan et al. (^{+}/Ca^{2+}-exchanger flux couple the membrane potential dynamics directly to the concentration dynamics.

b |
Total concentration of Troponin C (stationary buffer) | 53.0 μM |

b |
Total concentration of Calmodulin (mobile buffer) | 133.0 μM |

B |
Total concentration of SR buffer | 1500.0 μM |

b |
Total concentration of Fluo-4 | 133.0 μM |

k |
On rate for Troponin C binding | 0.043 μM^{−1} ms^{−1} |

k |
Off rate for Troponin C binding | 0.026 ms^{−1} |

k |
On rate for Calmodulin binding | 0.8 μM^{−1} ms^{−1} |

k |
Off rate for Calmodulin binding | 0.2 ms^{−1} |

k |
On rate for SR buffer binding | 0.1 μM^{−1} ms^{−1} |

k |
Off rate for SR buffer binding | 60.0 ms^{−1} |

k |
On rate for Fluo-4 binding | 0.0488 μM^{−1} ms^{−1} |

k |
Off rate for Fluo-4 binding | 0.0439 ms^{−1} |

D |
Diffusion constant of Troponin C | 0.04 μm^{2}/ms |

D_{B} |
Diffusion constant of SR buffer | 0.01 μm^{2}/ms |

D | Diffusion constant of calcium | 0.22 μm^{2}/ms |

D_{S} |
Diffusion constant of SR calcium | 0.20 μm^{2}/ms |

D_{Fluo−4} |
Diffusion constant of Fluo-4 | 0.033 μm^{2}/ms |

τ_{d} |
Refill-flux constant | 0.5 ms |

c_{0} |
Starting [Ca^{2+}]_{i} |
0.275 μM |

f_{NaCa, high} |
Maximal factor of g_{NaCa} at CRU centers |
2.5 |

f_{NaCa, low} |
Minimal factor of g_{NaCa} distant to CRUs |
1.5 |

f_{NaCa, surf} |
Factor of g_{NaCa} at surface |
0.5 |

K_{p} |
Uptake threshold | 0.2 μM |

Vp_{max} |
Strength of uptake | 0.8 μM/ms |

g_{NaCa} |
Strength of Na+/Ca^{2+}-exchanger |
0.84 μM/s |

k_{sat} |
Constant | 0.2 |

ξ | Constant | 0.35 |

K_{m, Nai} |
Constant | 12.3 mM |

K_{m, Nao} |
Constant | 87.5 mM |

K_{m, Cai} |
Constant | 0.0036 mM |

K_{m, Cao} |
Constant | 1.3 mM |

c_{naca} |
Constant | 0.3 μM |

C_{m} |
Cell capacitance | 3.1. × 10^{−4} μF |

ν_{cell} |
Whole cell volume | 2.58 × 10^{−5} μl |

Ω | Simulated sub-cell volume | 0.72 × 10^{−5} μl |

x × y | cross-section of simulated volume | 15 × 15 μm^{2} |

z | height of simulated volume | 32 μm |

F | Faraday constant | 96.5 C/mmol |

R | Universal gas constant | 8.315 Jmol^{−1} K^{−1} |

T | Temperature | 308 K |

External sodium concentration | 136 mM | |

Internal potassium concentration | 140 mM | |

External potassium concentration | 5.4 mM | |

_{ext} |
External calcium concentration | 1.8 mM |

ν_{sr}/ν_{cell} |
Ratio of SR to cell volume | 0.1 |

ν_{jsr}/ν_{cell} |
Ratio of jSR to cell volume | 0.005 |

ν_{cyt}/ν_{cell} |
Ratio of cytosolic volume to cell volume | 0.895 |

κ | Constant in Equation (20) | 1 nm^{−1} |

ϕ_{0} |
Constant in Equation (20) | −2.2 |

N_{c} |
Number of CRUs | 5120 |

N_{RyR} |
Average number of RyRs per CRU | 50 |

N_{LCC} |
Average number of LCCs per CRU | 12.5 |

r_{RyR, LCC} |
Ratio of RyRs and LCCs | 4.0 |

g_{LCC} |
LCC single channel conductance | 0.0546 μM^{3}/ms |

k |
Threshold for Ca-induced inactivation | 90.0 μM |

Threshold for Ca dependence of transition rate k_{6} |
60.0 μM | |

τ_{po} |
Time constant of activation | 1 ms |

r_{1} |
Opening rate | 0.3 ms^{−1} |

r_{2} |
Closing rate | 3 ms^{−1} |

s |
Inactivation rate | 0.00195 ms^{−1} |

k |
Inactivation rate | 0.00413 ms^{−1} |

k_{2} |
Inactivation rate | 0.0001 ms^{−1} |

k |
Inactivation rate | 0.00224 ms^{−1} |

T_{Ba} |
Time constant | 450 ms |

g_{RyR} |
RyR permeability | 2.33 μM^{3} s^{−1} |

k_{om} |
Activation rate | 60 s^{−1} |

k_{im} |
Inactivation rate | 5 s^{−1} |

k |
Maximal activation rate | 928.8 s^{−1} |

k |
Maximal inactivation rate | 7.8 s^{−1} |

K_{jsr} |
Half max. value for _{jsr}-effect on RyR |
550 μM |

K_{ac} |
Activation threshold | 8.5 μM |

K_{in} |
Inactivation threshold | 8.5 μM |

λ | Asymmetric inactivation | 27.5 |

g_{RyR} |
RyR permeability | 2.33 μM^{3} s^{−1} |

k_{open} |
Opening rate | 4.57 × 10^{5} ^{2.12} s^{−1} |

k |
Maximal opening rate | 800 s^{−1} |

k_{close} |
Closing rate | 245 ^{−0.27} s^{−1} |

g_{RyR} |
RyR permeability | 2.33 μM^{3} s^{−1} |

η | Ca^{2+} Hill coefficient |
2.1 |

k_{open} |
Opening rate | k^{+} ϕ^{η} |

k^{+} |
Opening rate constant | 1.107 × 10^{−4} ^{−1} μM ^{−η} |

k_{close} |
Closing rate | 0.2 s^{−1} |

ϕ | ||

ϕ_{k} |
1.5 mM | |

ϕ_{b} |
0.8025 |

g_{Na} |
Peak _{Na} conductance |
12.0 mS/μF |

g_{to, f} |
Peak _{to, f} conductance |
0.11 mS/μF |

g_{to, s} |
Peak _{to, s} conductance |
0.04 mS/μF |

g_{K1} |
Peak _{K1} conductance |
0.3 mS/μF |

g_{Kr} |
Peak _{Kr} conductance |
0.0125 mS/μF |

g_{Ks} |
Peak _{Ks} conductance |
0.1386 mS/μF |

g_{NaK} |
Peak _{NaK} conductance |
1.5 mS/μF |

The dynamics of the cytosolic Ca^{2+} concentration, ^{+}/Ca^{2+}-exchanger. The T-tubule network is an interface to extracellular fluid in the bulk of the cytosol due to which membrane fluxes like the Na^{+}/Ca^{2+}-exchanger contribute to bulk concentration dynamics (_{NaCa}). The Na^{+}/Ca^{2+}-exchanger flux through the plasma membrane (_{pump} describes the pumping of Ca^{2+} by SERCAs into the SR. A leak flux (often denoted by _{leak}) is not included as stochastic RyR openings during diastole account for SR leak. The Ca^{2+}-binding molecules (_{j}, _{cell}. The expressions for the fluxes are:

Ca^{2+} influx and release through LCC (

We use the bidomain concept (Keener and Sneyd, _{cyt}/ν_{cell} and ν_{sr}/ν_{cell}, resp. We include one buffer _{sr} in the SR lumen. The partial differential equation for the SR Ca^{2+} concentration ^{2+} buffer _{sr} are

_{S} and _{B} are diffusion matrices. Release appears as the flux _{jsr} in the

In this subsection we briefly introduce the membrane potential model of the rabbit ventricular myocyte by Mahajan et al. (

_{stim} is the stimulus current to depolarize the cell. _{ion} comprises (Mahajan et al.,

I_{Na} is the fast Na^{+} current, I_{K1} is the inward rectifier current, I_{to, f} is the fast component of the rapid outward K^{+} current, I_{to, s} is the slow component of the rapid outward K^{+} current, I_{Kr} is the rapid component of the delayed rectifier current, I_{Ks} is the slow component of the delayed rectifier current and I_{NaK} is Na^{+}/K^{+} pump current. These currents and the equations defining them are described in detail in Mahajan et al. (^{+}/Ca^{2+}-exchanger current _{NaCa} is determined by the spatial integral of the corresponding fluxes in the cytosolic concentration dynamics

A prefactor _{NaCa} for each finite element has been introduced to simulate the distribution of the Na^{+}/Ca^{2+}-exchanger according to Jayasinghe et al. (_{CaL} is the sum of all individual single channel currents (Equation 19) from the CRU models

The factor α = _{cell}/_{m}Ω converts the current in terms of ions/s into _{cell}. Similarly, the factor _{cell}/Γ in Equation (12) scales the plasma membrane component from simulated (Γ) to whole cell area (_{cell}). _{m} is the cell membrane capacitance and

The CRU model was based on a model previously developed by Schendel et al. (

The state dynamics of the LC- and RyR-channels are simulated with Markov models. For the RyRs we explored three models, the first is a model originally developed by Stern et al. (^{2+}concentrations. For an in-depth description of this model see (Schendel et al., ^{2+}dependence of the RyR closed time. With a small decline in jSR [Ca^{2+}] the Ca^{2+}flux via open RyRs declines, causing a decline in local dyadic [Ca^{2+}], which in turn causes a decrease in the open probability of neighboring RyRs, a process known as induction decay. This RyR model does not rely on experimentally un-substantiated biophysical mechanisms for CICR termination, such as dyadic/cytoplasmic RyR Ca^{2+}-dependent inactivation, or RyR-lumenal Ca^{2+}-dependent inactivation. The third model is the two-state model [see Figure ^{2+}] (^{−1} were not encountered.

_{o} is a fourth order Hill function of dyadic Ca^{2+}, the inactivation rate k_{i} is a first order Hill function of Ca^{2+}. The Ca^{2+} dependent rates are influenced by the Ca^{2+} concentration in the jSR. The model was originally developed by Stern et al. (_{open} and _{close} are polynomial functions of [Ca^{2+}]. _{open} is a polynomial function of both [Ca^{2+}] and [Ca^{2+}] _{jsr} (i.e., _{open} is influenced by the Ca^{2+}concentration in the jSR) and _{close} is a constant. _{Ca}, I2_{Ca} Ca^{2+} dependent inactivated states, C1, C2 closed states, I1_{Ba}, I2_{Ba} Ca^{2+} independent inactivated states (for details see Mahajan et al.,

For the LCCs we used the 7-state model developed by Mahajan et al. (^{2+} dependent and Ca^{2+} independent inactivation.

Each channel's state was chosen randomly according to the steady state distribution for the initial values. The transition times between different states were determined using the Gillespie Algorithm for time dependent transition rates. That algorithm determines the time of an event as the time of the crossing of a random threshold by an integrated propensity for each individual Markov chain. With the given rate of changes in membrane potential and Ca^{2+} concentration, propensities are strongly time dependent. Hence, before we finally accept a calculated update of

The frequency of transitions becomes very large with a large number of CRUs and with many channels in each CRU. While this in itself does not have a significant impact on the speed of the stochastic algorithm, it can force very small timesteps on the PDE model. In order to alleviate this problem and allow for longer timesteps we take advantage of the properties of the stochastic process. Those channel state transitions which do not represent either openings or closings are (within a given iteration step) independent stochastic events. We can therefore execute an arbitrary number of such transitions in a single iteration. This approach scales almost independently of the total number of states of the channel model. It is only dependent on the frequency of transitions from open to closed state or vice versa.

The Ca^{2+}concentration in the cleft (_{di}

Because the time to reach the stationary concentration profile upon opening or closing of a channel is about one order of magnitude shorter than the timescale of channel state transitions, we assume steady state for _{di}

The large aspect ratio radius/height of the dyadic space renders gradients in the

The values of ϕ_{0} and κ are listed in Table ^{1}_{i}, φ_{i})) we find
_{bulk} is the Ca^{2+}concentration at the boundary of the cleft (cylinder barrel) as computed by the PDE model.

With
_{di}

The currents ^{2+} concentration at the channel mouth:
_{jsr} and _{bulk}. The coefficients of that system can be calculated in advance of a simulation from the cleft geometry, which renders the simulation very efficient.

In order to employ the two state RyR-models by Cannell et al. (^{2+}buffering in the dyadic space. This is to be expected since in the original papers the models were fitted to data from experiments conducted in the presence of buffers such as Calmodulin, Fluo-4, and ATP. Cannell et al. (

We adjusted β so that the Ca^{2+} concentration at the channel mouths and the average dyadic Ca^{2+} concentrations matched the values given in the paper by Cannell et al. (^{2+}]_{i} that would be measured by a single-wavelength Fluo-4 experimental recording (denoted ^{2+}-bound Fluo-4. The _{d} is the dissociation constant of Fluo-4, ^{2+}-bound Fluo-4]), _{max} is the measured fluorescence intensity in Ca^{2+}-saturated dye (here this is set as the maximum [Ca^{2+}-bound Fluo-4], i.e., _{min} is the measured fluorescence intensity in the absence of Ca^{2+} (here set to zero) (see Table

Each dyadic space is paired with its own junctional SR (jSR) compartment. We assume spatially uniform Ca^{2+}concentration in the jSR. Ca^{2+}dynamics in the jSR depends on the release flux through the RyRs and a refill flux from the network SR (nSR), which we assume to depend simply on the concentration difference _{jsr}. The buffering by Calsequestrin is modeled using the fast buffer approximation.

^{2+}concentration at the location ^{−4} ms).

The modules interact on a time scale longer than a single iteration step by the dependencies of the dynamics on Ca^{2+}, ^{2+} currents and concentrations, between the CRU models and the electrophysiological model by the LCC current, and between the PDE system and the electrophysiological model via the bulk and plasma membrane components of the Na+/Ca^{2+}-exchanger flux (see also Figure

We use the Ca^{2+} concentration spatially resolved at the plasma membrane for calculating local values of _{NaCa}, and then average (Equation 12) to obtain the current entering the membrane potential dynamics. Vice versa, _{NaCa} as bulk source term for the PDEs.

The value of the Ca^{2+} concentration _{bulk} in Equation (17) to determine the RyR and LCC currents (

The membrane potential affects the LCC currents (Equation 19) and vice versa the sum of all individual LCC currents the membrane potential (Equation 13).

We used a piecewise bi-linear finite element method for the solution of the spatially three-dimensional reaction-diffusion model including the complex distribution of CRUs at multiple z-discs. The first challenge was the fine scale resolution of the computational grid to resolve the strong concentration gradients at the boundary of the CRUs. We take the equidistant tetrahedral elements with the size of 0.05 μM in our computations. The next challenge is to deal with the adaptive time stepping schemes for solving the reaction-diffusion systems. Due to the fast transitions of the channel opening/closings in a CRU, the time scales vary from tens of microseconds to milliseconds. To resolve such rapid changes adaptive and higher order time steppings are inevitable to treat the very smooth diffusion effects as efficiently as possible. To this end, we use higher order linearly implicit Runge-Kutta methods for time discretization of reaction-diffusion systems, see (Lang,

Our parallel implementation of the discretization routines are based on the public domain package DUNE (Bastian et al., ^{−6} is used as the stopping criteria for the linear solver at each step of the ODE time integrator.

Here we propose a novel technique to determine the new timestep during the stochastic opening of many CRUs. Due to the presence of the large numbers of CRUs, the stochastic algorithm that governs the timestep for the next channel transition plays an important role for the computations. As mentioned before, channel state transitions which do not represent openings or closings are, within a given iteration step, independent stochastic events. We can therefore execute several of them within a single iteration. Additionally, typical time steps during an AP are in the range of 0.01 ms, i.e., they are shorter than the diffusion time between neighboring CRUs. Consequently, conductance changing events in different CRUs are statistically independent on the time scale of a single iteration and we can allow for several of them in (distinct) CRUs within one time step.

We introduce two time steps: first the deterministic timestep τ_{det} (which is allowed by the numerical integration of the PDEs) and second the stochastic timestep τ_{stoc}. We propose the following algorithm.

The bulk calcium cycling PDE model and the electrophysiology model are integrated from _{det}, where τ_{det} is the accepted deterministic timestep of the PDE solver. Then, the stochastic channel transitions are predicted from _{det}. Suppose there were _{s} conductance changing stochastic events at times _{i} where _{s}, τ_{i} ≤ τ_{det} and 0 ≤ _{s} ≤ _{c}. Here, the time of the stochastic event is τ_{i} for the ^{th} CRU. In case that there is no stochastic event for a CRU, τ_{i} is set to τ_{i} = τ_{det}. The stochastic timestep τ_{stoc} is determined from the τ_{i} as the time by which a maximum number of acceptable transitions is reached. The maximum number has been determined empirically to be sufficiently small with 0.1 _{s} to cause no essential difference to simulations with τ_{stoc} sufficiently small to guarantee _{s} = 1. Now all the occurring events in the CRUs up to _{stoc} are set to take place at time _{stoc}. By doing so, we avoid time steps which are too small for acceptable simulation time. A schematic illustrating a single iteration and how the time steps are determined can be found in Figure

Our main result is the fully coupled simulation tool. Our motivation was to be able to take stochastic channel state dynamics for each channel in each CRU and the concentration profile within CRUs into account while executing the simulation of (partial) differential equations for other state variables. Figure ^{2+}-dependent transition rates than channels close to an open one.

^{2+} gradient in a CRU^{2+}] profile of a single open RyR is shown.

We simulated 16 z-discs, each with 320 CRUs, and the Ca^{2+} dynamics for this sub-cellular region (~30% of the cardiomyocyte) were coupled to the whole-cell electrophysiology ODE model. It took 64.2 h to solve a single action potential on 848 Intel Xeon E5-2650 v2 2.60 GHz CPUs (central processing units). The Ca^{2+} concentration profile 70.0 ms after stimulus at a single z-disc is shown in Figure ^{2+} dynamics and corresponding whole cell electrophysiology are shown in Figure ^{2+}] is visualized in Figure

^{2+}]_{i} and SR free [Ca^{2+}] at 70.0 ms after activation, using the Walker et al. (^{2+}]_{i}. ^{2+}] for the 8^{2+}]_{i} and nSR [Ca^{2+}] through an AP.

^{2+}] at 70.0 ms after activation, using the (Walker et al., ^{2+}]_{i} in green for [Ca^{2+}]_{i} = 0.6 μM and red for [Ca^{2+}]_{i} = 2.4 μM. ^{2+}] = 430 μM. There are 320 CRU per z-disc, amounting to 5160 CRUs in total, with an average of 50 RyR and 12.5 LCC per CRU. See Supplementary Movies ^{2+}]_{i} and nSR [Ca^{2+}] through an AP.

_{K1}, I_{Kr}, I_{Ks}, I_{NaK}, and I_{to, s}. _{CaL}), Na^{+}/Ca^{2+}-exchanger current (I_{NaCa}), I_{to, s}, and I_{Na} (truncated). ^{2+}]_{i} and ^{2+} -fluxes: J_{rel}, J_{Ca}, J_{NaCa}, J_{up}. ^{2+}] ([_{m}], [_{s}], [_{sr}]), and nSR lumenal [Ca^{2+}] ([Ca_{sr}]) are plotted. [_{m}] and [_{s}] are conventional concentrations in units of μM whereas [Ca_{sr}] and [_{sr}] are expressed for simplicity in units of μMν_{sr}/ν_{cyt} (μmol/l cytosol). Plots are of the first AP from a simulation with 16 z-discs.

The simulation shows that we can reproduce the whole sequence of Ca^{2+} transients from the initial concentration profile in the dyadic space, via sparks to the whole cell transient. The cytoplasmic [Ca^{2+}] is rather heterogeneous, due to randomness of release events as well as variations in CRU size. The concentration of 0.6 μM is reached in almost the whole volume, but 2.4 μM only in the proximity of the the z-discs. The currents reproduce basic features of the Mahajan-model (Mahajan et al., ^{2+} currents well (Weber et al., ^{+}/Ca^{2+}-exchanger current.

We explored the use of three RyR models which all produce realistic action potentials (Figure ^{2+} release and a low RyR maximum open percentage (2.7%). Gain (the ratio of _{rel} to _{Ca}) was ~4 during the first 50 ms of the AP and ~3 for the remainder of the AP (AP duration = 265 ms, basic cycle length BCL = 350 ms). With the (Cannell et al., ^{2+}]^{2+}]_{i} and a second [Ca^{2+}]_{i} rise and fall through the AP. With the (Walker et al., ^{2+} release than the (Stern et al., ^{2+}]_{i} transient with shorter BCL (Figure

^{2+}]_{i} and ^{2+}]_{i} and ^{2+}]_{i} and

^{2+}]_{i}].

This proof of concept study demonstrates a new multi-scale model of CICR linking three spatial scales: (1) detailed molecular stochastic modeling of the CRU at a continuous spatial scale; (2) a whole-cell ODE electrophysiology model, which describes the potassium channels, the voltage gated sodium channels, the sodium-potassium pump, and integrates all membrane fluxes to derive the total membrane current and voltage; (3) a PDE FEM calcium diffusion model representing myoplasmic and nSR Ca^{2+} diffusion between CRUs and between z-discs. One rationale for this approach is that it allows the removal of the artificial compartment for the sub-membrane space, and hence provides a more quantitative modeling of diffusion processes and gradients.

Taking gradients in the dyadic cleft into account in a simulation with thousands of CRUs would be impossible without the use of the stationary Green function. An estimate of the advantage of the quasi-static approximation with respect to computational speed can be obtained from estimating the number of operations per cleft. Our approach requires to solve a linear system of equations whenever channels open or close or the boundary condition or the jSR concentration has changed significantly. That means

To our knowledge, our model is the only whole cell AP model which includes CRUs clustered around the z-discs (a proxy for explicit T-tubules) ^{2+}] with realistic steep [Ca^{2+}] gradients. Other approaches include the multiscale model of Restrepo and Karma (^{2+} and mobile buffers, T-tubules, Markov chain models for the channels and excitation-contraction coupling gain. Cannell et al. (^{2+} transients and the activation of non-junctional RyR clusters of myocytes with sparse T-systems. A finite-element model has been implemented by Hatano et al. (^{2+}] in spatially resolved detail.

We explored the use of three RyR models: the 4-state model of Stern et al. (^{2+}] (^{2+}]_{i} equals 0.8 μM with time to peak of 200 ms. With the (Cannell et al., ^{2+}]_{i} were observed with an early peak of [Ca^{2+}]_{i} = 0.6 μM at time to peak ~10 ms, with the principal peak [Ca^{2+}]_{i} transient of ~0.8 μM and time to peak 100 ms. With the (Walker et al., ^{2+}]_{i} was ~1.0 μM and time to peak 130 ms. The time to peak [Ca^{2+}]_{i} for both the 2-state models is comparable to that reported for the rabbit by Weber et al. (^{2+}]_{i} local maximum with the (Cannell et al., ^{2+} sparks and blinks, and has never been previously used in a model of prolonged CICR resulting from an AP. Indeed the authors of this model state that the model equations are not designed to capture behavior at resting [Ca^{2+}]_{i} with fully loaded SR. The RyR-dynamics of this model with its current parameters are therefore reproducing spark like behavior in an AP. We had to introduce strong buffering inside the dyadic space and jSR-depletion in order to reach closing rates sufficiently fast for termination of release at the end of an action potential. However, this proof of concept investigation does not provide evidence that induction decay alone is an insufficient mechanism to cause the termination of CICR in an AP. Rather, the model provides a platform for investigating AP CICR. Careful experimentally led tuning of the 2-state model parameters alongside Ca^{2+} buffering parameters and the jSR refill flux will allow assessment of the feasibility of induction decay as the sole CICR termination mechanism in an AP. A second possible explanation for the early local maximum in [Ca^{2+}]_{i} with the (Cannell et al., ^{2+}]_{i} uptake and release by the mitochondria. Although mitochondrial [Ca^{2+}]_{i} uptake and release is controversial on such fast timescales (Boyman et al., ^{2+}]_{i} uptake can act like a fast stationary buffer, and that inhibition of fast uptake through a specific inhibitor of the mitochondrial Ca^{2+} uniporter (MCU) can result in an early local maximum of the [Ca^{2+}]_{i} transient (Maack et al., ^{2+}]_{i} recorded in experiments via a fluorescent Ca^{2+} probe such as Fluo-4 [approximated as (^{2+}]_{i} peak, if this were a genuine experimental feature of cardiomyocyte CICR. In AP simulations with the 2-state RyR models, some local high calcium transients at CRU-sites remained into the diastolic period, associated with CRUs where some RyRs remained open. A similar phenomenon has been described in the setting of spontaneous Ca^{2+}-sparks by Stern et al. (^{2+}] (

The discrepancies between our simulations and the ODE-Mahajan model with respect to Na^{+}/Ca^{2+}-exchanger current illustrate the value of spatially resolved modeling in exploring detailed properties of Ca^{2+} dynamics. Using the same Na^{+}/Ca^{2+}-exchanger model as Mahajan et al. (^{+}/Ca^{2+}-exchanger current. In both models the current has the same general profile, with a short period in reverse mode (positive current) in the first phase of an action potential, followed by forward mode (negative current) during the _{NaCa} current turns negative much later during the plateau than in the ODE model. The Na^{+}/Ca^{2+}-exchanger depends on the [Ca^{2+}] in sub-membrane space in the ODE model and on the cytoplasmic concentration ^{2+}] in the ODE model exhibits a sharp rise and decline while the average myoplasmic concentration shows slower dynamics. We have experimented with localizing the Na^{+}/Ca^{2+}-exchanger molecule density in proximity to the CRUs to reflect the observed higher abundance of Na^{+}/Ca^{2+}-exchanger near the CRU (Scriven et al., ^{+}/Ca^{2+}-exchanger current through investigation of these problems.

We showed on a level of proof-of-concept that multiscale modeling of cardiomyocyte ECC from sub-dyadic scales to many z-discs using full partial differential equations is possible. We demonstrate that the model produces realistic physiology on these scales and has the potential to provide new insight into subcellular mechanisms and structures. Maybe the most severe limitation of this modeling approach is the requirement for high performance computing (HPC) to run it. We expect some improvement of simulation efficiency by more specifically tailored numerical methods, however the requirement for HPC will remain. The use of the Green function inside the dyadic cleft requires linearity of the reaction diffusion equations there. That excludes non-linear buffering terms and allows for linear buffering only. Some of the limitations of our model arise from its early state of development and will be removed with inclusion of more detail like e.g., anisotropic diffusion, mitochondria, more detailed buffering, and non-junctional RyR. We used modular programming as far as possible to ease the exploration of a variety of ion channel models and other species specific membrane potential dynamics.

NC implemented the PDE model and coupled this with the CRU model and cell electrophysiology ODE model. JV co-implemented the cell electrophysiology ODE and PDE model. WN and SG implemented the RyR Markov-models. WN implemented the CRU model and co-implemented the PDE model. NC, WN, JV, SG carried out simulations, processed data from simulations and produced results figures. SG, NC, WN, JV, MF drafted the manuscript. MF conceived the CRU model, the PDE model and the coupling of the model scales. All authors approved and assisted in redrafting the final manuscript.

This study has been supported by DFG grant FA350/9-1, DFG GRK 1772, BMBF grant eMed:SMART and a collaborative grant from the German Centre for Cardiovascular Research to MF and S. Luther, MPI DS Göttingen.

The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

We acknowledge the work of Thomas Schendel who developed the CRU model which was adapted to the CRU model used in this study (Schendel et al.,

The Supplementary Material for this article can be found online at:

^{2+}] by means of 0.6 μM and 2.4 μM free myoplasmic [Ca^{2+}]_{i} isosurfaces; ^{2+}]_{i} transient; ^{2+}].

_{sr}] by means of a 430 μM isosurface; ^{2+}]_{i} transient; _{sr}].

^{2+}current amplitude and open channel probability.

^{2+}]

_{i}transients in guinea pig ventricular myocytes.

^{2+}release sites in rat cardiac myocytes.

^{2+}ions in the dyadic cleft; continuous versus random walk description of diffusion.

^{2+}release unit in the ventricular cardiac myocyte.

^{2+}channels?

^{2+}channel and Na

^{+}/Ca

^{2+}exchange localization in cardiac myocytes.

^{2+}channels on early after-depolarizations.

_{3}receptor channel clusters and concentration profiles.

^{1}A factor of