Search tips
Search criteria 


Logo of nihpaAbout Author manuscriptsSubmit a manuscriptHHS Public Access; Author Manuscript; Accepted for publication in peer reviewed journal;
IEEE Eng Med Biol Mag. Author manuscript; available in PMC 2011 October 19.
Published in final edited form as:
PMCID: PMC3197248

Multi-Scale Modeling in Rodent Ventricular Myocytes: Contributions of structural and functional heterogeneities to excitation-contraction coupling


There is a growing body of experimental evidence suggesting that the Ca2+ signaling in ventricular myocytes is characterized by high gradient near the cell membrane and a more uniform Ca2+ distribution in the cell interior [1]–[7]. An important reason for this phenomenon might be that in these cells the t-tubular system forms a network of extracellular space, extending deep into the cell interior. This allows the electrical signal, that propagates rapidly along the cell membrane, to reach the vicinity of the sarcoplasmic reticulum (SR) where intracellular Ca2+ required for myofilament activation is stored [1], [8]–[11]. Early studies of cardiac muscle showed that the t-tubules are found at intervals of ~2 μm along the longitudinal cell axis in close proximity to the Z-disks of the sarcomeres [12]. Subsequent studies have demonstrated that the t-tubular system has also longitudinal extensions [9]–[11], [13].

The SR is an entirely intracellular, membrane-bounded compartment that abuts but is not continuous with the sarcolemma. The junctions where SR approaches the sarcolemma contain specialized proteins [1], [4], [14]. The sarcolemmal L-type Ca2+ channels (LCC) are located primarily at the SR junctions where the Ca2+ release channels in the SR, the ryanodine receptors (RyRs), exist [2], [3], [11], [15]–[17]. The RyRs are arranged in organized arrays of hundreds of receptors up to 200 nm in diameter.

The concept that the LCCs and RyRs form a local functional unit (release unit, RU) is supported by the observations of Ca2+ sparks. Ca2+ sparks reflect the nearly synchronous activation of a cluster of about 6–20 RyRs at a single junction. Ca2+ sparks are the fundamental units of the SR Ca2+ release both at rest and during cell excitation [1], [3]–[7]. Thus, the microanatomy of t-tubules and SR permits spatially homogeneous and synchronized SR Ca2+ release throughout the cell. During physiologically normal excitation-contraction coupling (EC-coupling) a several thousand Ca2+ sparks in each cell are synchronized in time by the action potential to achieve a spatially homogeneous Ca2+ transient [1], [4]–[6], [8], [14]. It has also been observed that the spatially uniform Ca2+ transient might be achieved if the SR Ca2+ release and uptake are abolished [8]. However the mechanisms underlying cell activation synchrony and Ca2+ homogeneous distribution still remains unclear.

Recent immunohistochemical studies but one [17] have demonstrated also that marked variations in the distribution of Ca2+-handling proteins (L-type Ca2+ channel, Na+/Ca2+ exchanger, sarcolemmal Ca2+ ATPase) along the cell membrane probably exist [10], [11], [15], [16]. The analysis suggests that most of the L-type Ca2+ channels are concentrated in the t-tubules (from 3 to 9 times more in the t-tubule membrane than on the surface sarcolemma) and that the concentration of LCC along the t-tubule increases toward the center of the cell [10], [16].

Studies on the distribution of the main Ca2+ efflux pathway, the Na+/Ca2+ exchanger (NCX), are more controversial. All studies but one [18] have reported NCX to localize both to the surface and t-tubule membrane, and most studies suggest that the NCX is 1.7 to 3.5 times more concentrated in the t-tubule membrane, [15], [18]–[20]. However, Kieval et al. data [21] indicate the NCX is more evenly distributed. The distribution of the sarcolemmal Ca2+ ATPase is also unclear [10]. Only one study reports that in hamster and canine ventricular cells this Ca2+ efflux pathway is located predominantly in the surface membrane [22]. In summary, the observed differences in the spatial distribution and molecular architecture of Ca2+ microdomains suggest that significant differences in the EC-coupling between the cell surface and cell interior may exist. However how the localization of Ca2+-handling proteins along the sarcolemma regulates the intracellular Ca2+ signaling still remains uncertain.

Taken together above studies demonstrate that remarkable amount of fundamental quantitative data on the ventricular cell structure and function has been accumulated. Recently it has been also emphasized that biophysically realistic computational models, incorporating transverse-axial t-tubular system and considering geometric irregularities and inhomogeneities in the distribution of ion-transporting proteins, are missing and needed [11], [23]. For this reason, our main goal here was to develop a detailed 3-D model at the sub-cellular level that would allow us to examine how the distribution of Ca2+ fluxes via t-tubule and surface membrane may affect Ca2+-entry, diffusion and buffering. Thus, SR Ca2+ uptake and release was not included here. The current model of the rat ventricular myocyte includes: (1) a simplified 3-D geometry of a single t-tubule and its surrounding half-sarcomeres; (2) the spatially distributed L-type Ca2+ channel, Na+/Ca2+ exchanger, sarcolemmal Ca2+ pump and background Ca2+ leak along the sarcolemma; and (3) the stationary and mobile endogenous Ca2+ buffers (troponin C, ATP, calmodulin) and the exogenous mobile Ca2+ buffer, Fluo-3.

The results suggest that, in the presence of 100 μM Fluo-3, the model is able to predict a uniform Ca2+ distribution inside the cell if Ca2+ microdomains are distributed heterogeneously along the cell membrane. In the absence of Ca2+ indicator the model predicts non-uniform Ca2+ distribution in the cytoplasm and high Ca2+ gradient near the cell edges when the Ca2+ flux pathways were distributed heterogeneously. We concluded that the distribution of Ca2+ handling proteins along the cell membrane might be another important mechanism regulating ventricular EC-coupling. These model predictions are in qualitative agreement with published experimental data in rat ventricular myocytes [8]. Preliminary results of this work have been presented to the Biophysical Society in abstract form [24].

It is important to mention here that this 3-D sub-cellular model of single t-tubule and surrounding structures also yields insights across two other scales of biological organization: a microscopic scale of individual Ca2+ RU and a whole-cell scale. It allows us not only to extend the analysis further to integrate models of individual Ca2+ RU, SR Ca2+ pump or leak but also to examine how experimentally suggested spatial distributions of these Ca2+ transporters [2], [3], [11], [15]–[17] may affect the behavior of a single RU or the mechanisms underling a synchronized SR Ca2+ release. Future modeling efforts will be focused on replacing the idealistic t-tubule and surrounding structures geometries with more realistic or to include several surrounding t-tubules and other sub-cellular organelles that might help to understand better normal and pathophysiological mechanisms.

Materials and Methods

Model cell geometry and spatial protein distribution along the cell membrane

The model geometry was derived from the published structural data [9], [11], [23]. The cell model contains one repeating unit inside the ventricular myocyte, including single t-tubule and its surrounding half-sarcomeres (Fig. 1A and Table 1). The t-tubule was assumed to be cylinder, with a diameter of 0.25 μm and a depth of 6.87 μm. The surrounding half-sarcomeres were modeled as a cube-shaped box enclosing the t-tubule with dimensions 2 μm ×2 μm in the plane of the cell surface and 7 μm in depth. The volume of the model compartment was estimated to be ~27.68 μm3. The compartment membrane area was ~9.34μm2 where the percentage of cell membrane within t-tubule was 58% (~5.42 μm2) and within the surface membrane 42% (~3.92 μm2), [9], [11], [23], [25].

Fig. 1
Panel A: Schematic drawing of the model geometry showing the single t-tubule and its surrounding half-sarcomeres. The top surface of the cube is the surface membrane for the model compartment. Panel B: A diagram illustrating Ca2+ entrance and extrusion ...
Table 1
Physical constants and cell geometry

The accessible volume for Ca2+ was estimated from reported measurements in adult rat ventricular myocytes [1], [26]. These data, suggest that myofilaments occupy 47–48 % of the cell volume, mitochondria 34–36%, nucleus 0–2%, t-tubule system 0–1.2% and SR lumen 3.5%. The experiments also suggest that about 50% of the myofilament space is accessible for Ca2+ ions (i.e. contains water) and that mitochondria and nuclei are not rapidly accessible for Ca2+ [1], [27]. In this study we also assume that the SR lumen is not accessible to Ca2+ in the presence of ryanodine and thapsigargin. Thus, from the experimental data and above assumptions, the accessible volume for Ca2+ in adult rat ventricular cells was estimated to be ~35–37% of the total cytosolic volume (Vacc) ~12.9–13.6 pL.

The depth of the cleft between the sarcolemmal and SR membranes, where the LCC and RyR localize, has been reported to be 12–20 nm [27] and the sub-sarcolemma space between cellular membrane and myofibrils ~45 nm [1]. In this study, the size of both spaces was considered infinitely thin on the scale of the continuum model. This assumption allowed: (1) not to explicitly define a different diffusion coefficients for Ca2+ and mobile buffers (Fluo-3, ATP, Cal) in the cleft, sub-sarcolemmal and myofibril spaces as in Michailova et al. [27], (2) to simplify the model and to improve the model stability and efficiency.

In agreement with reported experimental data, Ca2+ transporters were distributed heterogeneously along the model cell surface (Fig. 1B and and2A).2A). The concentration of LCC in the t-tubule membrane was 6 times of that in the surface membrane and increased 1.7-fold along the length of the t-tubule. The concentration of NCX in the t-tubule membrane was three times that in the surface membrane. The sarcolemmal Ca2+ ATPase was located only in the surface membrane. The background Ca2+ leak was assumed to be present throughout the whole sarcolemma because no data were available of how this Ca2+ channel is distributed.

Fig. 2
Panel A: The distribution of L-type Ca2+ current was computed (dashed line) by multiplying the experimentally measured cluster density and fluorescent intensity plots (solid line), [8], [16]. Panel B: The L-type Ca2+ current density was fitted and plotted ...

In the axial t-tubule direction, the distribution of LCC current was computed by combining the cluster density and fluorescent intensity plots [16]. The data were then scaled and fitted by a cubic polynomial:


where: x is the distance from the cell surface.

The parameter values of the polynomial (pj, j=1–4) are shown in Table 2. This polynomial was further scaled by a single factor C (see Table 2) such that the total Ca2+ flux along the t-tubule membrane remained unchanged by redistributing the Ca2+ fluxes.

Table 2
The parameter values of cubic polynomial describing the L-type Ca2+ current distribution along the t-tubule

Reaction-diffusion equations

The effects of four exogenous and endogenous Ca2+ buffers (Fluo-3, ATP, calmodulin, troponin C) were considered (Fig. 1B). The endogenous stationary buffer troponin C (TN) was distributed uniformly throughout the cytosol but not on the cell membrane. The free Ca2+ and mobile buffers (Fluo-3, ATP, calmodulin) diffuse and react throughout the cytoplasm and cell membrane subject to reflective boundary conditions at the cell surfaces. The nonlinear reaction-diffusion equations describing Ca2+ and buffers dynamics inside the cell are:






where: [Bm] mobile buffer Fluo-3, calmodulin or ATP; [Bs] stationary buffer troponin C.

The diffusion coefficients for Ca2+, CaATP, CaCal and CaFluo as well as the total buffer concentrations and rate buffer constants used in the model are shown in Table 3. In the model we also assume: (1) Ca2+ binds to Fluo-3, calmodulin, ATP, and TN without cooperativity; (2) the initial total concentrations of the mobile buffers are spatially uniform; (3) the diffusion coefficients of Fluo-3, ATP or calmodulin with bound Ca2+ are equal to the diffusion coefficients of free Fluo-3, ATP or calmodulin.

Table 3
Ca2+ and buffer reaction-diffusion parameters

The total Ca2+ flux (JCaflux) throughout the t-tubule and surface membrane is:


where: JCa - total LCC Ca2+ influx; JNCX - total NCX Ca2+ efflux; JpCa - total Ca2+ pump efflux; JCab - total background Ca2+ leak influx.

To fit the whole-cell LCC current density to reported data in rat myocytes with SR release inhibited [28] we used a MATLAB implementation of Hinch et al. model [29], (Fig. 2B). To describe the Na+/Ca2+ exchanger, membrane Ca2+ pump and leak current densities we used expressions from Hinch et al.:




Flux parameter values were estimated or taken from the literature (see Table 4). At rest the Ca2+ influx via background Ca2+ leak was adjusted to match the Ca2+ efflux via NCX and Ca2+ ATPase so that no net movement across the cell membrane occurred.

Table 4
Membrane Ca2+ fluxes parameters

In the model, each current density (Ii) was converted to Ca2+ flux (Ji) by using the experimentally suggested surface to volume ratio ( CmVcell8.8pF/pL) in adult rat ventricular myocytes [1], [25]:


Then, the total compartment Ca2+ flux (JCaflux) was computed by multiplying each total Ji with the model cell volume (Vmc), and distributing JCaflux to the surface membrane and t-tubule membrane according to the prescribed Ca2+ handling protein concentration ratio.


The nonlinear reaction diffusion system was solved using a finite difference method in time and finite element method in space. Here we developed a distributed finite element software package, to be used on a Linux cluster for parallel computations. Computations took ~18 minutes to simulate 400 ms of one Ca2+ cycle on 10 processors of an Intel Xeon-based cluster (see Fig 3B). A discrete time stepping of 4ms was used during the simulations.

Fig. 3
Panel A: The finite element software package contained four components with data flowing from top to bottom. Panel B: Composite view of the subdomain–to-processor assignment used to partition the mesh on 10 processors.

The software was built using several established software packages. Mesh generators included Netgen4.3.1 and TetGen1.3 authored by Schöberl and Si respectively [30]. The parallel finite element assembler was based on libMesh0.4.3-rc2 by Kirk et al. [31]. The discrete linear system was solved using PETSc-2.2.1 [32]. An operator splitting method was used to de-couple the reaction-diffusion system [33]. The nonlinear ordinary differential equations were solved using an A-stable and implicit Runge-Kutta method of order 5. The simulation results were visualized using GMV3.4 [34]. Post-processing and data analysis were performed using customized Python and MATLAB version 7.1 (The MathWorks, Natick, MA) scripts.

In designing the software, the portability and reusability were emphasized over computational efficiency. The implementation was based on object-oriented programming. Therefore the resulting software contains a set of loosely connected application tools, which focused on both flexibility and functionality. The structure of the software followed that of a finite element simulation package (Fig. 3A).


Ca2+ concentration changes in the presence of 100 μM Fluo-3 and inhibited SR release and uptake

Model results in Fig. 4 were computed for conditions approximating those of experiments by Cheng et al. [8] (see Fig. 4M), who examined Ca2+ signals in rat ventricular myocytes in the presence of 100 μM Fluo-3 and pharmacological blockade of the SR. The voltage-clamp protocol (holding potential −50mV, electric pulse of 10mV for 70ms) and whole-cell L-type Ca2+ current were derived from the study of Zahradnikova et al. [28] with blocked SR (see Fig. 2B and Figs. 4A–B).

Fig. 4
Model predictions in the presence of 100 μM Fluo-3. The voltage-clamp protocol and the whole-cell L-type Ca2+ current used in this set of simulations are shown in panels A–B. Predicted global Na+/Ca2+, Ca2+ pump and leak currents and global ...

Consistent with the experimental study [8] in the model the scanned line was located at 200nm from the surface of the t-tubule (see red line in Fig. 5A). The line-scan images and local Ca2+ time-courses are shown in Figs. 4G–I and Figs. 4J–L, respectively. These results suggest that the model was able to predict uniform Ca2+ distribution inside the cell when LCC and NCX current densities were heterogeneously and Ca2+ leak homogeneously distributed along the sarcolemma and Ca2+ pump was located on the surface membrane. Furthermore, when the LCC flux density was 6 times higher and uniform in the t-tubule, the Ca2+ concentration profiles were less evenly distributed (Figs. 4H and 4K) but the predicted variations in [Ca2+]i seem to be within the range of experimental noise in Fig. 4M. Finally, these studies revealed that heterogeneous Ca2+ transients might occur if the LCC current density was uniformly distributed throughout the whole cell surface (Figs. 4I and 4L). The model also demonstrated that distributing NCX flux homogeneously along the sarcolemma did not significantly affect Ca2+ uniformity (data not shown). The distribution of Ca2+ pump and leak pathways also did not seem to have a significant effect. Blockade of these fluxes reduced the control [Ca2+]i peak in Fig. 4J by less than 0.4% (data not shown). Figures 4F and 4C–E show the global [Ca2+]i transient, Na+/Ca2+exchanger, Ca2+ pump and leak currents when Ca2+ pathways were distributed heterogeneously as in Fig. 4G. Figures 4F and 4J–L illustrate: (1) that the global and all local Ca2+ transients reached the peak after ~ 68 ms; and (2) that the decay of the Ca2+ signals was extremely slow, since the reuptake of Ca2+ to the SR was blocked and [Ca2+]i was lowered solely by extrusion from the cell via the Na+-Ca2+ exchanger and Ca2+ pump. Possible reasons for the predicted extremely slow NCX rate might be that in the model the intracellular Na+ concentration ([Na+]i) was kept constant (in contrast to existing evidence for the presence of local sub-membrane [Na+]i gradients on the action potential time-scale [1], [35]) or that the realistic distribution of NCX flux density probably differs as assumed in the model. The 3-D Ca2+ concentration distributions at Ca2+ peak (line scan 200 nm away from the t-tubule) and the local Ca2+ transients (line scan 0 nm and 875 nm away from the t-tubule) are shown in Fig. 5.

Fig. 5
Predicted 3-D Ca2+ concentration distributions (computed from the line-scan images in Figs. 4G–I) at Ca2+ peak of 68 ms are shown in panels A–C. In panel D the spatial profiles at Ca2+ peak along the scanning line (200nm from the surface ...

In summary, the results in Fig. 4 and Fig. 5 suggest that in the presence of 100 μM Fluo-3 and with the SR blocked (1) Ca2+ concentration near the surface membrane decreased while [Ca2+]i in the cell interior increased when Ca2+ transporters were uniformly distributed and after that heterogeneously redistributed, and (2) in each moment of the cell cycle the overall Ca2+ distribution remained almost uniform across the model compartment when Ca2+ transporters were heterogeneously distributed.

The good agreement between the model and experimental observations suggests that heterogeneous channel distributions may be another important mechanism regulating intracellular Ca2+ distribution and myofilament function. Moreover, it allows the model to be used to simulate experiments that cannot be performed because of technical reasons (for example in the absence of Ca2+ indicator).

Ca2+ concentration changes in the absence of Fluo-3, inhibited SR and heterogeneous distribution of Ca2+ membrane pathways

Figure 6 shows membrane currents, and Ca2+ signals arising from the ionic influx via L-type Ca2+ channels at zero Fluo-3 with heterogeneously distributed membrane Ca2+ fluxes (as in Fig. 4G) during voltage-clamp stimulation.

Fig. 6
Model predictions in the absence of Fluo-3 with heterogeneous distribution of Ca2+ pathways via the cell membrane. Panels A and B show the voltage-clamp protocol and whole-cell L-type Ca2+ current used in this set of simulations. The predicted global ...

Since it has been suggested [2], [36] that the dye does not affect Ca2+ entry via L-type channels, the same global LCC flux was used during this numerical experiment (Fig. 4B and Fig. 6B). In the absence of Fluo-3 the peak of average Ca2+ transient increased ~1.6-fold (from 0.19 μM up to 0.3 μM), while the time to peak (~ 68 ms) remained unchanged (Fig. 6). Figures 6H and 6I–J demonstrate that local Ca2+ peaks also increased while the times to peak remained ~ 68 ms. In addition, [Ca2+]i decay at zero Fluo-3 was slow. The increase in local Ca2+ levels across the cell membrane affected Na2+/Ca2+ exchange and Ca2+ pump activities more significantly than the Ca2+ background leak (see Figs. 6C–E and Figs. 4C–E). It is interesting that, under these conditions, the model predicts nonuniform Ca2+ concentration distributions in and along the cell membrane and in the cell interior. Figures 6I–K demonstrate that in the absence of Fluo-3 [Ca2+]i in and near the t-tubule membrane was much more higher than in the cell interior. These results indicate that Ca2+ levels vary in the transverse direction too. Along the t-tubule, [Ca2+]i was higher near to the surface membrane and in the cytoplasmic t-tubule end while in the cell interior [Ca2+]i was higher only near the surface membrane.


In this study we developed a 3-D continuum model of Ca2+-signaling, buffering and diffusion inside a small representative region of the rat ventricular muscle cell. The simplified model geometry, derived from published structural data [9], [11], [23], contained one repeating unit inside the myocyte, including single cylindrical t-tubule and its surrounding half-sarcomeres. Following the morphological studies on 3-D reconstruction of t-tubule system, ~58% of membrane surface was assumed to be within the t-tubule. On the basis of experimental data in rat myocytes [26] the aqueous sub-cellular volume, accessible to Ca2+, was estimated to be ~ 35–37%. In addition, we assumed the cleft and sub-membrane sizes infinitely thin that allowed us not to explicitly define a deferent values for DCa, DCaFluo, DCaATP and DCaCal in these near-membrane spaces and in myofibrils. To test the correctness of above assumption we examined how 2-fold increase in DCa or mobile buffer diffusion coefficients (values suggested in water) in the cleft and sub-membrane space or variations in the sub-membrane depth (from 12 nm to 45 nm) would affect local and global Ca2+ transients. We found that these changes had insignificant effects on the calculated Ca2+ signals (data not shown). We concluded that a functional importance of these spaces lies in the immediate vicinity of the sarcolemma where all membrane proteins and sarcolemmal ion channels are located [17], [27]. In view of the fact that Ca2+ signaling in cells is largely governed by Ca2+ diffusion and binding to mobile and stationary Ca2+ buffers [3], [27], [37] the effect of four Ca2+ buffers (Fluo-3, ATP, calmodulin, TN) was considered. We validated the model against published experimental data on Ca2+ influx, membrane protein distributions and Ca2+ diffusion in rat ventricular myocytes treated with ryanodine and thapsigargin [8], [10], [15], [16], [18]–[22]. We used the model: (1) to examine how the distribution of L-type Ca2+ channels, Na+/Ca2+ exchangers, membrane Ca2+ ATPases and membrane leaks regulates the spatio-temporal features of intracellular Ca2+ signals in the presence of fluorescent dye; and (2) to simulate and analyze the Ca2+ signals that are not accessible experimentally, i.e. in the absence of Ca2+ indicator. The parallel numerical software enabled us to solve the reaction-diffusion equations in a reasonable time and to test the model carefully.

An important model limitation is that we simplified the t-tubule and surrounding structures geometries assuming them cylindrical and cube-shaped box, respectively. However, several studies provide evidence that in normal ventricular cells the realistic t-tubule geometry is quite complex (with large local variations in the diameter and transverse-axial anatomies) and that its surrounding structures might form quite arbitrary shapes (see Fig. 7) [9], [13], [38]. It has been also observed that in the failing hearts the t-tubules might be aberrantly shaped or dilated [13], [39]. Taken together above data strongly suggest that further extending of the current model toward more realistic geometries is needed. The use of idealistic shapes might change the diffusion distances in plane and depth directions and consequently the predicted Ca2+ distributions.

Fig. 7
T-tubular network complexity in adult mammalian ventricular myoyctes. Cardiac sarcolemma including t-tubules was stained with Alexa-488-conjugated wheat germ agglutinin (WGA) in 80 μm vivratome sections of adult mouse ventricular myocardium and ...

Ca2+ signals in the presence of 100 μM Fluo-3 and absence of SR activity

Cheng et al. [8] examined the propagation of EC-coupling in isolated rat ventricular cells by using laser scanning confocal microscope and 100 μM Fluo-3. They found that the depolarization-evoked [Ca2+]i transients in the presence or absence of SR release and uptake are activated synchronously near the cell surface and in the cell interior and that the time of [Ca2+]i rise does not depend on whether SR activity is abolished or not. They concluded that the lack of systematic differences in the Ca2+ fluorescence signal recorded from either the center or the edge of the cell indicates that sarcolemmal Ca2+ and the SR Ca2+-release channels (RyRs) are distributed throughout the heart cell, and that ventricular EC-coupling is not limited by diffusion of the second messenger from the surface of the cell to the center.

In this study, to investigate further the mechanisms underlying EC-coupling propagation in rat ventricular myocytes, we used modeling approach. We examined how the distribution of the sarcolemmal Ca2+ influx and efflux transporters regulates Ca2+ movement from the cell surface to the cell interior when the SR was blocked. In agreement with experiment [10], [16], we found that in the presence of 100 μM Fluo-3 model predicts a homogenous Ca2+ distribution inside the cell if L-type Ca2+ current density is ~6 times higher in the t-tubule than in surface membrane and increases ~1.7 fold along the t-tubule length. An interesting model observation was also that the uniform Ca2+ distribution might be achieved assuming LCC flux density 6-fold higher and uniform along the t-tubule because the predicted [Ca2+]i fluctuations here were within the range of experimental noise [8]. New experiments should be performed to test this hypothesis.

In addition, our results revealed that the spatio-temporal features of local Ca2+ signals depended on the diffusion distances in the axial and cell surface directions. Thus, when the LCC were distributed uniformly the local Ca2+ peak in radial depth (7 μm) decreased from ~0.25 μM to ~0.15 μM while in the other cell directions (1 μm × 1 μm) no significant changes were found. Redistributing the amount of Ca2+ pumped via the cell membrane (i.e. increasing LCC current density ~6-fold along the t-tubule) while keeping total Ca2+ flux unchanged, lowered Ca2+ gradients near the surface membrane and increased Ca2+ levels in the cell interior.

Other interesting model findings were that: (1) the global Ca2+ time-course and time to [Ca2+]i peak (~68 ms) do not depend on whether Ca2+ flux pathways are distributed homogeneously, uniformly in the t-tubule or heterogeneously (data not shown); (2) the changes in the local Ca2+ transients near the cell membrane when Ca2+ microdomains were distributed differently affected NCX and Ca2+ pump time-courses while Ca2+ leak current remained unchanged (data not shown); and (3) the NCX, Ca2+ pump or Ca2+ leak redistribution alone were not able to alter significantly the predicted Ca2+ uniformity (data not shown).

Ca2+ signals in the absence of Fluo-3 and SR activity and heterogeneous distribution of Ca2+ membrane fluxes

Another advantage of the model was its ability to predict Ca2+ signals that would occur in the absence of Fluo-3. The model simulations revealed that at zero Fluo-3 and with 260 μM ATP and 24 μM calmodulin as mobile buffers, the Ca2+ distribution in the cell interior would be non-uniform if LCC are distributed heterogeneously (as in Figs. 4G). Note in the absence of 100 μM mobile Fluo-3 a local and global Ca2+ peaks increased while the time of Ca2+ rise remained almost unchanged. Furthermore, during the Ca2+ influx larger, steeper and heterogeneous Ca2+ concentration gradients were predicted between the cytosol and sub-membrane space while in the cell interior [Ca2+]i was more uniformly distributed. In addition, results suggest that the calculated sub-membrane [Ca2+]i levels were higher: (1) near the t-tubule mouth because a close topological proximity of this membrane to the surface sarcolemma additionally increased the relative amount of Ca2+ entering there; (2) near the cytosolic t-tubule end because the LCC flux density was assumed higher. The simulations also showed that removing Fluo-3 affected NCX and Ca2+ pump time-courses by increasing local free [Ca2+]i. Thus, these findings clearly reveal that (1) the exogenous Fluo-3 may act as a significant buffer and carrier for Ca2+, and that (2) the use of 100 μM Fluo-3 during the experiment may sensitively alter the realistic Ca2+ distribution. However the question arising here is: Based on the model what might be the underlying mechanism(s) for the predicted Ca2+ concentrations gradients in the absence of Fluo-3? A reasonable answer is that Ca2+ movement and distribution inside the cell also strongly relies on the presence of mobile and stationary Ca2+ buffers [27], [37]. Thus now in the absence of Fluo-3 (1) the stationary Ca2+ buffer troponin C (TN) imposed stronger diffusion barrier for Ca2+ that resulted in larger and steeper Ca2+ concentration gradients between the cytosol and narrow sub-membrane space, (2) the buffer capacity and spread of ATP and Cal alone via the t-tubule membrane were not able to mask the assumed heterogeneous Ca2+ entering via the sarcolemma, (3) in the cell interior, the overall Ca2+ distribution remained almost uniform in each moment of the cell cycle (because TN, ATP and Cal buffering capacity dominated) and a little beat higher in and near the surface membrane (because [TN] was zero there).

Taken together, our studies suggest that in ventricular myocytes when the SR is pharmacologically inhibited: (1) intracellular Ca2+ concentration rapidly increases during Ca2+ entrance (0–70 ms) while the decay of [Ca2+]i is slow; (2) in the absence of fluorescent dye, large Ca2+ concentration gradients might develop near the cell membrane; and (3) intracellular Ca2+ distribution is tightly regulated by the localization of Ca2+ transporter proteins along the sarcolemma and strongly relays on the presence of mobile and stationary Ca2+ buffers. These studies also imply that in ventricular cells with intact and functional SR, the Ca2+ signal most likely would spread faster along the t-tubule and surface membrane than to the cell interior and that in the absence of Ca2+ dye high Ca2+ gradients under the surface membrane and more uniform Ca2+ distribution in the cell interior might be expected.


The authors thank Ernst Niggli (University of Bern), Tomas Shannon (Rush University), and J. Zhang (JHU) for the valuable suggestions and discussions. We also thank the reviewers of the manuscript for useful suggestions.


This work was supported by NSF grant BES 0506252 (McCulloch), the National Biomedical Computational Resource (NIH grant 2 P41 RR08605) and Burroughs Wellcome Foundation (Postdoctoral Fellowship for Dr. Lu). Work in the McCammon group is supported by NIH, NSF, HHMI, and the Center for Theoretical Biological Physics.


1. Bers DM. Excitation-contraction coupling and cardiac contractile force. Kluwer Academic Press; Dortrecht, Boston, London: 2001.
2. Soeller C, Cannell MB. Analysis of ryanodine receptor clusters in rat and human cardiac myocytes. PNAS. 2007;104(38):14958–14963. [PubMed]
3. Izu LT, Means SA, Shadid JN, Chen-Izu Ye, Balke CW. Interplay of Ryanodine Receptor Distribution and Calcium Dynamics. Biophys J. 2006;91:95–112. [PubMed]
4. Berridge MJ, Bootman MD, Roderick HL. Calcium signaling: dynamics, homeostasis and remodeling. Nat Rev Mol Cell Biol. 2003;4:517–529. [PubMed]
5. Niggli E. Ca2+ sparks in cardiac muscle: Is there life without them? News Physiol Sci. 1999;14:129–134. [PubMed]
6. Blatter LA, Kockskamper J, Sheehan KA, Zima AV, Huser J, Lipsius S. Local calcium gradients during excitation-contraction coupling and alternans in atrial myocytes. J Physiol. 2003;546:19–31. [PubMed]
7. Sobie EA, Dilly KW, Cruz JS, Lederer WJ, Jafri MS. Termination of cardiac Ca2+ sparks: An investigative mathematical model of calcium-induced calcium release. Biophys J. 2002;83:59–78. [PubMed]
8. Cheng H, Cannell MB, Lederer WJ. Propagation of excitation-contraction coupling into ventricular myocytes. Pflugers Arch. 1994;428:415–417. [PubMed]
9. Soeller C, Cannell MB. Examination of the transverse tubular system in living cardiac rat myocytes by 2-photon microscopy and digital image – processing techniques. Circ Res. 1999;84:266–275. [PubMed]
10. Brette F, Orchard C. T-tubule function in mammalian cardiac myocytes. Circ Res. 2003;92:1182–1192. [PubMed]
11. Pásek M, Brette F, Nelson A, Pearce C, Qaiser A, Christe G, Orchard CH. Quantification of T-tubule area and protein distribution in rat cardiac ventricular myocytes. Prog Biophys Mol Biol. 2008;96:244–257. [PubMed]
12. Page E, McCallister LP. Quantitative electron microscopic description of heart muscle cells: application to normal, hypertrophied and thyroxin-stimulated hearts. Am J Cardiol. 1973;31:172–181. [PubMed]
13. Martone ME, Edelman VM, Thor A, Young SJ, Lamont SP, Ross J, Jr, Ellisman MH. Three dimensional analysis of transverse tubules in normal and failing heart: A combined confocal and high voltage electron microscope study. Microscopy and Microanalysis. 1997;3(S2):231–232.
14. Bers DM. Calcium cycling and signaling in cardiac myocytes. Annu Rev Physiol. 2008;70:23–49. [PubMed]
15. Scriven DR, Dan P, Moore EW. Distribution of proteins implicated in excitation-contraction coupling in rat ventricular myocytes. Biophys J. 2000;79:2682–2691. [PubMed]
16. Scriven DR, Klimek A, Lee KL, Moore EW. The molecular architecture of calcium microdomains in rat cardiomyocytes. Ann NY Acad Sci. 2002;976:488–499. [PubMed]
17. Scriven DR, Klimek A, Asghari P, Bellve K, Moore EW. Caveolin-3 is adjacent to a group of extradyadic ryanodine receptors. Biophys J. 2005;89:1893–1901. [PubMed]
18. Yang Z, Pascarel C, Steele DS, Komukai K, Brette F, Orchard CH. Na+-Ca2+ exchange activity is localized in the T-tubules of rat ventricular myocytes. Circ Res. 2002;91(4):315–22. [PubMed]
19. Despa S, Brette F, Orchard CH, Bers DM. Na/Ca exchange and Na/K-ATPase function are equally concentrated in transverse tubules of rat ventricular myocytes. Biophys J. 2003;85(5):3388–3396. [PubMed]
20. Thomas MJ, Sjaastad I, Andersen K, Helm PJ, Wasserstrom JA, Sejersted OM, Ottersen OP. Localization and function of the Na+/Ca2+-exchanger in normal and detubulated rat cardiomyocytes. J Mol Cell Cardiol. 2003;35(11):1325–1337. [PubMed]
21. Kieval RS, Bloch RJ, Lindenmayer GE, Ambesi A, Lederer WJ. Immunofluorescence localization of the Na-Ca exchanger in heart cells. Am J Physiol. 1992;263:C545–C550. [PubMed]
22. Iwata Y, Hanada H, Takahashi M, Shigekawa M. Ca(2+)-ATPase distributes differently in cardiac sarcolemma than dihydropyridine receptor alpha 1 subunit and Na+/Ca2+ exchanger. FEBS Lett. 1994;355(1):65–68. [PubMed]
23. Pásek M, Šimurda J, Christé G, Orchard CH. Modelling the cardiac transverse-axial tubular system. Prog Biophys Mol Biol. 2008;96:226–243. [PubMed]
24. Lu S, Holst MJ, Bank RE, McCulloch AD, Michailova AP. 3-D model of synchronous calcium signaling in ventricular myocyte. Biophys J. 2005;88:136A.
25. Satoh H, Delbridge LM, Blatter LA, Bers DM. Surface:volume relationship in cardiac myocytes studied with confocal microscopy and membrane capacitance measurements: species-dependence and developmental effects. Biophys J. 1996;70(3):1494–14504. [PubMed]
26. Page E. Quantitative ultrastructural analysis in cardiac membrane physiology. Am J Physiol. 1978;235:C146–C158. [PubMed]
27. Michailova A, DelPrincipe F, Egger M, Niggli E. Spatiotemporal features of Ca2+ signaling, buffering and diffusion in atrial myocytes with inhibited sarcoplasmic reticulum. Biophys J. 2002;83:3134–3151. [PubMed]
28. Zahradnikova A, Kubalova Z, Pavelková J, Györke S, Zahradník I. Activation of calcium release assessed by calcium release-induced inactivation of calcium current in rat cardiac myocytes. Am J Physiol Cell Physiol. 2004;286(2):C330–C341. [PubMed]
29. Hinch R, Greenstein JL, Tanskanen AJ, Xu L, Winslow RL. A simplified local control model of calcium induced calcium release in cardiac ventricular myocytes. Biophys J. 2004;87:3723–3736. [PubMed]
30. Schöberl J. NETGEN - automatic mesh generator.
31. Kirk BS, Peterson JW, Stogner R, Petersen S. The libMesh library.
32. Balay S, Bushelman K, Eijkhout V, Gropp WD, Kaushik D, Knepley MG, McInnes LC, Smith BF, Zhang H. Tech Rep ANL-95/11. Argonne National Laboratory; 2004. PETSc user’s manual.
33. Sundnes J, Lines GT, Tveito A. Efficient solution of ordinary differential equations modeling electrical activity in cardiac cells. Math Biosci. 2001;172(2):55–72. [PubMed]
34. Ortega F. The General Mesh Viewer.
35. Bers DM, Barry WH, Despa S. Intracellular Na+ regulation in cardiac myocytes. Cardiovasc Res. 2003;57:897–912. [PubMed]
36. Grantham CJ, Cannell MB. Ca2+ influx during the cardiac action potential in guinea pig ventricular myocytes. Circ Res. 1996;79:194–200. [PubMed]
37. Zhou Z, Neher E. Mobile and immobile calcium buffers in bovine adrenal chromaffin cells. J Physiol. 1993;469:245–273. [PubMed]
38. Sachse FB, Savio-Galimberti E, Goldhaber JI, Bridge JHB. Sub-micrometer anatomical models of the sarcolemma of cardiac myocytes based on confocal imaging. Pacific Symposium Biocomputing. 2008;13:390–401. [PubMed]
39. Louch WE, Bito V, Heinzel FR, Macianskiene R, Vanhaecke J, Flameng W, Mubagwa K, Sipido KR. Reduced synchrony of Ca2+ release with loss of T-tubules-a comparison to Ca2+ release in human failing cardiomyocytes. Cardiovasc Res. 2004;62:63–73. [PubMed]