Conceived and designed the experiments: APM. Performed the experiments: YC. Analyzed the data: YC ADM APM. Contributed reagents/materials/analysis tools: YC ZY MH MJH JAM APM. Wrote the paper: YC JAM APM.
The authors have declared that no competing interests exist.
The t-tubules of mammalian ventricular myocytes are invaginations of the cell membrane that occur at each Z-line. These invaginations branch within the cell to form a complex network that allows rapid propagation of the electrical signal, and hence synchronous rise of intracellular calcium (Ca2+). To investigate how the t-tubule microanatomy and the distribution of membrane Ca2+ flux affect cardiac excitation-contraction coupling we developed a 3-D continuum model of Ca2+ signaling, buffering and diffusion in rat ventricular myocytes. The transverse-axial t-tubule geometry was derived from light microscopy structural data. To solve the nonlinear reaction-diffusion system we extended SMOL software tool (
In cardiac muscle cells, calcium (Ca2+) is best known for its role in contraction activation. A remarkable amount of quantitative data on cardiac cell structure, ion-transporting protein distributions and intracellular Ca2+ dynamics has been accumulated. Various alterations in the protein distributions or cell ultra-structure are now recognized to be the primary mechanisms of cardiac dysfunction in a diverse range of common pathologies including cardiac arrhythmias and hypertrophy. Using a 3-D computational model, incorporating more realistic transverse-axial t-tubule geometry and considering geometric irregularities and inhomogeneities in the distribution of ion-transporting proteins, we analyze several important spatial and temporal features of Ca2+ signaling in rat ventricular myocytes. This study demonstrates that the computational models could serve as powerful tools for prediction and analyses of how the Ca2+ dynamics and cardiac excitation-contraction coupling are regulated under normal conditions or certain pathologies. The use of computational and mathematical approaches will help also to better understand aspects of cell functions that are not currently amenable to experimental investigation.
Ventricular cardiac muscle cells have deep invaginations of the extracellular space known as t-tubules
In cardiac muscle cells, several computational models have been introduced to investigate the Ca2+ signaling, buffering and diffusion
In the present study, we sought to develop a morphologically correct geometric model of the t-tubule and to use this model for computational studies of the intracellular Ca2+ dynamics. We examined the Ca2+ signaling in rat ventricular myocytes that had been treated with ryanodine and thapsigargin to eliminate Ca2+ release and uptake by the SR. By using published electro-physiological data and laser-scanning confocal [Ca2+]i measurements, we were able to analyze several important spatial and temporal features of the Ca2+ signals in these cells. In this context, our goal was at least three-fold. The first aim was to develop a mathematical model that would be in qualitative or quantitative agreement with published experimental measurements on Ca2+ influx, and Ca2+ buffering and diffusion in rat ventricular cells with SR function inhibited
Combining light microscopy (LM) and electron microscopy (EM) together with 3-D tomographic reconstruction, Hayashi
(A,
The extreme intricacy of the t-tubule system in mice (with transverse-axial anatomies and large local variations in t-tubule diameter) has been observed in rat ventricular myocytes as well
Symbol | Definition | Value | Ref. |
F | Faraday constant | 96.5 C mmol−1 | Physical constant |
T | Temperature | 295 K | Physical constant |
R | Universal gas constant | 8.314 J mol−1 K−1 | Physical constant |
Cell volume | 36.8 pL | ||
Cell capacitance | 324 pF | ||
Accessible volume for Ca2+ | 12.9–13.6 pL | Estimated | |
Compartment volume | 23.31 µm3 | Estimated | |
Compartment surface | 9.00 µm2 | Estimated | |
T-tubule radius | 0.19–0.469 µm | ||
T-tubule depth | 5.645 µm | ||
Cell surface direction | 2µm | ||
Cell surface direction | 2µm | ||
Depth | 5.96µm |
Recent immunohistochemical studies have demonstrated that marked variations in the distribution of Ca2+-transporting protein complexes (L-type Ca2+ channel, Na+/Ca2+ exchanger) along the cell membrane probably exist
Studies on the distribution of the main Ca2+ efflux pathway, the Na+/Ca2+ exchanger, are more controversial. All studies but one
In the current model, the effects of four exogenous and endogenous Ca2+ buffers (Fluo-3, ATP, calmodulin, troponin C) were considered (
The diffusion coefficients for Ca2+, CaATP, CaCal and CaFluo as well as the total buffer concentrations and buffer rate constants used in the model are shown in
Symbol | Definition | Value | Ref. |
Extracellular Ca2+ concentration | 1000 µM | ||
Resting Ca2+ concentration | 0.1 µM | ||
Total Fluo-3 concentration | 100 µM | ||
Total free ATP concentration | 260 µM | ||
Total troponin concentration | 70 µM | ||
Total calmodulin concentration | 24 µM | ||
Diffusion coefficient for Ca2+ | 0.39 µm2 ms−1 | ||
Diffusion coefficient for CaFluo | 0.1 µm2 ms−1 | ||
Diffusion coefficient for CaATP | 0.168 µm2 ms−1 | ||
Diffusion coefficient for CaCal | 0.025 µm2 ms−1 | ||
Ca2+ on-rate constant for TN | 0.04 µM−1 ms−1 | ||
Ca2+ off-rate constant for TN | 0.04 ms−1 | ||
Ca2+ dissociation constant for TN | 1 µM | ||
Ca2+ on-rate constant for CaATP | 0.225 µM−1 ms−1 | ||
Ca2+ off-rate constant for CaATP | 45 ms−1 | ||
Ca2+ dissociation constant for ATP | 200 µM | ||
Ca2+ on-rate constant for CaFluo | 0.23 µM−1 ms−1 | ||
Ca2+ off-rate constant for CaFluo | 0.17 ms−1 | ||
Ca2+ dissociation constant for Fluo | 0.739 µM | ||
Ca2+ on-rate constant for Cal | 0.125 µM−1 ms−1 | ||
Ca2+ off-rate constant for Cal | 0.2975 ms−1 | ||
Ca2+ dissociation constant for Cal | 2.38 µM |
The total Ca2+ flux (
To describe L-type Ca2+ current, Na+/Ca2+ exchanger, Ca2+ leak current densities we used the following expressions:
Flux parameter values were estimated or taken from the literature (see
Symbol | Definition | Value | Ref. |
Constant | 1 | ||
Constant | 4 ms | ||
Constant | 70 ms | ||
Extracellular Na+ concentration | 140 mM | ||
Resting Na+ concentration | 10 mM | ||
Pump rate of NCX | 38.5 µM ms−1 | ||
Voltage dependence of NCX control | 0.35 | ||
Na+ half saturation of NCX | 87.5 mM | ||
Ca2+ half saturation of NCX | 1380 µM | ||
Low potential saturation factor of NCX | 0.1 | ||
Conductance | 3.4e-6µM mV−1ms−1 | Estimated | |
Conductance | −6.8e-6µM mV−1ms−1 | Estimated |
In the model, each current density (Ii) was converted to Ca2+ flux (Ji) by using the experimentally suggested surface to volume ratio (
The voltage-clamp protocol (holding potential −50mV, electric pulse of 10mV for 70ms) and whole-cell L-type Ca2+ current were derived from Zahradnikova
In finite element methods, a complex domain needs to be discretized into a number of small elements (such as triangles or tetrahedra). This process is usually referred to as mesh generation
The nonlinear reaction diffusion system was solved by a finite difference method in time and finite element method in space using our SMOL software tool (Smoluchowski Solver,
In agreement with the reported experimental data
(A) The distribution of L-type Ca2+ current is computed (
The parameter values of the polynomial (pj, j = 1–4) are shown in
Symbol | C | p1 | p2 | p3 | p4 |
0.4515 | −4.1379e-4 | −1.1722e-2 | 1.978e-1 | 1.0033 |
Consistent with the Cheng
Diagrams illustrating external membrane, t-tubule mouth, and positions of scanning line (
Model results in
(A–B) The voltage-clamp protocol and whole-cell L-type Ca2+ current. (C–E) Predicted global Na+/Ca2+ and Ca2+ leak currents and global Ca2+ transient when no detectible differences in [Ca2+]i are found (see panel F). (F–H) Calcium concentrations visualized as line-scan images in transverse cell direction. (I–K) Local Ca2+ transients taken at three featured spots along the scanning line of interest: 0.17 µm –
(A–C) 3-D views of the Ca2+ concentration distribution at Ca2+ peak of 76 ms. In (D) the spatial profiles at Ca2+ peak along the scanning line of interest are compared. L-type Ca2+ flux density heterogeneously distributed along the t-tubule membrane (A and
These results demonstrate that with LCC heterogeneous or LCC six times higher in the t-tubule: (1) predicted Ca2+ concentration profiles were non-uniform when t<100 ms but the variations in [Ca2+]i seem to be within the range of experimental noise in
Predicted global [Ca2+]i transient, Na+/Ca2+exchanger, and Ca2+ leak currents with LCC pathways heterogeneously distributed (as in
This model is also able to predict how the Ca2+ transients are regulated at different line-scan positions within this geometrically irregular micro-domain. Note, due to the technical limitations the Cheng
The 3-D Ca2+ concentration distributions and spatial Ca2+ profiles at Ca2+ peak (76 ms) are shown in
Additional interesting model findings are that: (1) large and steep [Ca2+]i gradients were predicted inside the sub-sarcolemmal 3-D space (see
In this study the value of 390 µm2 s−1 for diffusion coefficient of free Ca2+ and published buffer diffusion coefficients and parameters were used to compare the calculated Ca2+ signals with the Cheng's et al. fluorescence Ca2+ signals recorded in rats
We could not find experimental data suggesting
During simulations of SR Ca2+ release into the diadic cleft, a major effect of the stationary phospholipids Ca2+ binding sites has been suggested
(A–B) The voltage-clamp protocol and whole-cell L-type Ca2+ current used in this set of simulations. (C–D) The predicted global Na+/Ca2+ and Ca2+ leak currents. (E–F) The global Ca2+ transient and Ca2+ concentrations visualized as line-scan image in the transverse cell direction. (G) Local Ca2+ transients taken at three featured spots along the scanning line (0.17 µm –
The model predicts here that the spread and buffer capacity of 100 µM Fluo-3 were able to mask completely the pronounced spatial non-uniformities in [Ca2+]i distribution that will occur during the Ca2+ influx when the SR Ca2+ metabolism is inhibited
The conjecture made in the present model, that the endogenous calmodulin and ATP are mobile Ca2+ buffers, allowed us to examine how the mobility of these buffers would affect the Ca2+ dynamics and membrane flux time-courses within this irregular micro-domain.
(A–B) The voltage-clamp protocol and whole-cell L-type Ca2+ current. (C–E) Quantitative comparison of the effects of buffer mobility on the global Na+/Ca2+ and Ca2+ leak currents and global Ca2+ transient (
In this study, we also examined the effects of NCX inhibition on the voltage-clamp induced Ca2+ signals in the absence of fluorescent indicator. The inhibition of NCX forward mode was achieved by removing extracellular sodium (i.e. 0 mM [Na+]e). To adjust Ca2+ flux via Ca2+ leak to match Ca2+ influx via NCX, so that at rest no net movement across the cell membrane to occur, we estimated Ca2+ leak constant (
(A–B) The voltage-clamp protocol and whole-cell L-type Ca2+ current. (C) Quantitative comparison of the effects of changes in [Na+]e on the global Na+/Ca2+ flux (
The current study attacks a difficult problem on how to incorporate the structural-based biological information, critical for the subcellular and cellular function, into sophisticated computational investigations. Pursuing this goal we developed a 3-D continuum model of Ca2+ signaling in rat ventricular cells that incorporates the realistic transverse-axial t-tubule topology and considers geometric irregularities and inhomogeneities in the distribution of ion-transporting proteins. The t-tubule micro-architecture was extracted from the Hayashi
The model was validated against published experimental data on Ca2+ influx, membrane protein distributions and Ca2+ diffusion in rat cells treated with ryanodine and thapsigargin to inhibit the SR Ca2+ metabolism
The model studies with 100 µM Fluo-3 indicate also that the [Ca2+]i gradients depend 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 (5.96 µm) decreased from ∼1.5 fold 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 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 (see
It should be noted, that in our previous work we used the simplified t-tubule geometry (assuming cylindrical shape) to simulate the Ca2+ dynamics in rats
A surprising and important finding of this study is that the spread and buffer capacity of 100 µM Fluo-3 were able to mask completely the pronounced spatial [Ca2+]i non-uniformities that would occur during the Ca2+ influx in the absence of dye (see
Taken together, these studies provide compelling evidence that (1) the exogenous Fluo-3 acts as a significant buffer and carrier for Ca2+, and that (2) the use of 100 µM Fluo-3 during the experiment can sensitively alter the realistic Ca2+ distribution. A new the question, however, arises: Based on the above model findings what could be the underlying mechanism(s) for the predicted heterogeneous Ca2+ concentrations gradients in the absence of Fluo-3? A reasonable answer is that the Ca2+ movement and distribution inside the cell rely strongly not only on the specific cell micro-architecture and Ca2+ transporters distribution but also on the presence of endogenous mobile and stationary Ca2+ buffers (ATP, calmodulin, troponin C - known to affect strikingly the Ca2+ dynamics)
Important limitations of the current modeling approach are: (1) the relatively small size of the model compartment that contains only a single realistic t-tubule shape and spans by just a half-sarcomere inside the ventricular myocyte; and (2) the assumption that the model compartment is a repeating unit inside the cell. The structural studies, however, provide evidence that in rodent ventricular myocytes the realistic t-tubule network is quite complex, (see
Important limitation of this study is also that we assume that the ion flux pathways are continuously distributed throughout the t-tubule membrane. Immunohistochemical studies, however, suggest that L-type Ca2+ channels appear to be concentrated as discrete clusters in the dyadic clefts (narrow spaces between LCC and RyR) distributed regularly along the t-tubule membrane at relatively small distances of ∼0.68 µm,
Finally, in the present model the effects of two endogenous Ca2+ mobile buffers (ATP, calmodulin) and one stationary Ca2+ buffer (troponin C) were considered only. The ventricular cells, however, contain other stationary Ca2+ buffers (including phospholipids, myosin, calsequestrin) or small and high mobile Ca2+ binding molecules (ADP, AMP) that were not included in the model (or may be other stationary and mobile buffers that have not been identified yet),
Simulations presented in this study demonstrate that the more accurate knowledge of transverse-axial t-tubule microanatomy and protein distributions along the sarcolemma is important to better understand the mechanisms regulating the excitation-contraction coupling in rat ventricular myocytes. The results demonstrate that Ca2+ movement from the cell membrane to the cell interior relies also strongly on the presence of mobile and stationary Ca2+ buffers, including the Ca2+ indicator dye. The key findings are: (1) the model predicts lack of detectible differences in the fluorescence Ca2+ signals at the peripheral and deep myoplams when the membrane Ca2+ flux is heterogeneously distributed along the sarcolemma; (2) 100 µM mobile Fluo-3 is able to mask the pronounced spatial non-uniformities in the [Ca2+]i distribution that would occur when the SR Ca2+ metabolism is inhibited; (3) during the Ca2+ influx alone, large and steep Ca2+ gradients are predicted in the narrow sub-sarcolemmal space (∼40–50 nm in depth); (4) in rodents the branched t-tubule topology, the punctuate spatial distribution of Ca2+ flux along the sarcolemma and the endogenous Ca2+ buffers actually function to inhibit Ca2+ waves. Improved functional and structural computational models are needed to guide the experiments and to test further our understanding of how the t-tubule microanatomy and protein distributions regulate the normal cell function or cell cycle under certain pathologies. To our best knowledge, this study is the first attempt to use the finite element methods to investigate the intracellular Ca2+ responses in physiologically realistic transverse-axial t-tubule geometry.
100 µM Fluo-3: Calcium flux heterogeneously distributed (
(5.45 MB AVI)
Zero Fluo-3 and calcium flux heterogeneously distributed: ATP and calmodulin mobile (
(4.05 MB AVI)
Zero Fluo-3 and calcium flux heterogeneously distributed: Extracellular sodium 140 mM (
(4.07 MB AVI)
The authors thank Shaoying Lu (Urbana-Champaign, University of Illinois) for the valuable advices and discussions and Yuhan Fu for technical assistance. We also thank the reviewers of the manuscript for useful suggestions. The SMOL and FETK source codes are available to download from