1. Computational and Systems Biology
  2. Physics of Living Systems
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The Ca2+ transient as a feedback sensor controlling cardiomyocyte ionic conductances in mouse populations

  1. Colin M Rees
  2. Jun-Hai Yang
  3. Marc Santolini
  4. Aldons J Lusis
  5. James N Weiss
  6. Alain Karma  Is a corresponding author
  1. Northeastern University, United states
  2. Northeastern University, United States
  3. Cardiovascular Research Laboratory, David Geffen School of Medicine, University of California, United states
  4. David Geffen School of Medicine, University of California, United States
Research Article
Cite this article as: eLife 2018;7:e36717 doi: 10.7554/eLife.36717
7 figures, 6 tables, 1 data set and 1 additional file


Schematic representation of the sarcolemmal currents and intracellular Ca2+ cycling proteins of the mouse ventricular myocyte model.
Effects of individual conductances on the Ca2+ transient (CaT).

(A) CaT amplitude defined as the difference Δ[Ca]i between the peak and diastolic values of the cytosolic Ca2+ concentration [Ca]i versus G/Gref where G is the individual conductance value and Gref some fixed reference value. (B) Time-averaged [Ca]i over one pacing period ([Ca]i) versus G/Gref. Illustration of the effect of varying ICa,L conductance (C) and Ito,f conductance (D) on AP and CaT profiles, where 50%, 100%, and 150% correspond to Gref=0.5, 1.0, and 1.5, respectively. (E) Effect of varying RyR conductance on SR Ca2+concentration [Ca]SR and CaT. Different time windows are plotted for the CaT and SR load (0 to 150 ms) and AP waveforms (0 to 50 ms) in (C-–E).

Figure 3 with 3 supplements
Computationally determined good enough solutions (GES) with calcium sensing.

(A) Examples of GES representing combinations of 6 conductances that produce a normal CaT and intracellular Na+ concentration. Each color represents a different GES and the corresponding AP and CaT profiles are shown in B) and C), respectively. (D) Histograms of individual normalized conductances G/Gref for a collection of 7263 GES showing that some conductances are highly variable while others are highly constrained. (E) Three-dimensional (3D) plot revealing a three-way compensation between conductances of ICa,L, Ito,f, and IKur. Each GES is represented by a red dot. All GES lie close to a 2D surface in this 3D plot. Pairwise projections (grey shadows) do not show evidence of two-way compensation between pairs of conductances. (F) Alternate representation of three-way compensation obtained by plotting ICa,L versus the sum of Ito,f and IKur. Peak values of those currents after a voltage step from −50 to 0 mV are used to make this plot that can be readily compared to experiment. Different time windows are plotted for the AP waveforms and CaT in B and C, respectively.

Figure 3—figure supplement 1
Histograms of individual ion channel conductances in 8320 GESs found by a GES search constrained only by Ca2+ transient amplitude and average, but not constrained by intracellular sodium concentration [Na]i.
Figure 3—figure supplement 2
Correlation between ICa,L and the sum of Ito,f and IKur is weaker but still significant when intracellular sodium concentration is not constrained.
Figure 3—figure supplement 3
Computationally determined GES with voltage sensing.

(A) Examples of GES representing combinations of 6 conductances that produce a normal AP with predominantly voltage sensing. Sensors are S1 = APD90, S2 = APD30 and S3 = [Na]i with ϵ=0.05. Each color represents a different GES and the corresponding AP and CaT profiles are shown in (B) and (C), respectively. Correlation between ICa,L and the sum of Ito,f and IKur is not present among models constrained by APD90 (D) or by APD90, APD30 and [Na]i (E).

Figure 4 with 1 supplement
Good enough solutions in the Hybrid Mouse Diversity Panel (HMDP).

(A) Central result of this paper showing quantitative agreement between theoretically predicted and experimentally measured compensation of inward Ca2+ and outward K+ currents. Equivalent plot of Figure 3F showing the sum of Ito,f and IKur versus ICa,L for nine different mouse strains using peak values of those currents (proportional to conductances) after a voltage step from −50 to 0 mV. Mean current values (green filled squares) are shown together with standard errors of the mean (thin bars) for each strain. The number of cells used for each strain is given in Table 1 of the Materials and methods section. Computationally determined GES are superimposed and shown as faded red points using all three sensors (CaT amplitude, average [Ca]i, and diastolic [Na]i) and faded blue points for two sensors (CaT amplitude and average [Ca]i). Lines represent linear regression fits using the method of Chi-squared minimization with errors in both coordinates including (solid line, p=0.0144) and excluding (dashed line, p=0.0007) the outlier strain BXA12/PgnJ marked by a red box. The small p values of those fit validate the computationally predicted three-way compensation of Ca2+ and K+ currents. The three strains selected for the organ scale study (C57BL/6J, CXB1/ByJ, and BXA25/PgnJ) with low, medium, and high ICa,L conductance, respectively, are highlighted by blue circles. (B) Cell shortening, measured as the fraction of resting cell length at 4 Hz pacing frequency in different HMPD strains where thick and thin bars correspond to standard error of the mean and standard deviation, respectively. A standard ANOVA test shows no significant differences in cell shortening between strains (p=0.4136) supporting the hypothesis that different combinations of conductances produce a similar CaT and contractile activity.

Figure 4—figure supplement 1
Patch clamp measurements of mean ICa,L (A), Ito,f (B), and IKur (C) functional current density averaged over multiple cells for nine HMDP mouse strains with standard errors (thick bars) and standard deviations (thin bars).
Figure 5 with 2 supplements
Organ scale compensation.

(A) Mean ICa,L conductance in three different HMDP strains where thick and thin bars denote standard error and standard deviation, respectively. (B) Sets of conductances generated to be representative of individual cells within ventricular tissue of the three strains by assigning normally distributed random values to the ICa,L, Ito,f and IKur conductances using experimentally determined means and standard deviations. The blue, green, and red points correspond to the three HMDP strains with low (C57BL/6J), medium (CXB1/ByJ), and high (BXA25/PgnJ) ICa,L conductance, respectively, and the grey points are the results of the three-sensor GES search (same as Figure 3F). (C) Variable AP waveforms in uncoupled myocytes with conductances randomly chosen from the distribution shown in B for C57BL/6J and D) AP waveforms for coupled myocytes in tissue for C57BL/6J and the two other strains. AP waveforms of uncoupled cells vary significantly from cell to cell as observed experimentally (Fig. Figure 5—figure supplement 1) but are uniform in electrotonically coupled cells, as expected. (E) Histograms of Ca2+ transient (CaT) amplitude (ΔCa) and action potential duration (APD) for C57BL/6J in electrotonically uncoupled and coupled cells. Importantly, in coupled cells, the more uniform APD translates into a much more uniform CaT amplitude, reflecting the strong effect of the cell’s APD on its CaT amplitude. (F) Distribution of CaT amplitudes within electrotonically coupled cells in tissue scale simulations using the parameter distributions from B. The three strains have the same mean CaT amplitude averaged over all cells marked by a thick vertical gray line, thereby demonstrating that compensation of Ca2+ and K+ currents remains operative at a tissue scale. (G) Distribution of CaT amplitudes obtained by varying only ICa,L conductance and with Ito,f and IKur conductances fixed to their reference values. Lack of compensation between Ca2+ and K+ currents in this case yields different mean CaT amplitude.

Figure 5—figure supplement 1
Action potential recordings from isolated myocytes for mouse strain C57BL/6J paced at 4 Hz under current clamp.

The recordings illustrate the typical degree of cell-to-cell variability of AP morphology observed in all strains.

Figure 5—figure supplement 2
Histogram of average Ca2+ concentration corresponding to Figure 5E for C57BL/6J in electrotonically uncoupled and coupled cells.
Correlation between L-type Ca2+ current conductance and cardiac hypertrophic response to a stressor for different HMDP strains.

The Pearson correlation is r = 0.86 (p=3e-4).

Compensation and gene expression.

Plot showing the existence of a statistically very significant correlation (Pearson correlation coefficient r=0.47 and p-value, p=8.11013) between the expression level of Kcnip2, encoding the KChIP2 accessory β subunits that interact with Kv4.2 channels (Ito,f) and of Cacna1c, a gene encoding the α1C subunit of the Cav1.2 L-type calcium channels (ICa,L) across 206 mice. Cardiac gene expression was measured in 106 control (Pre-ISO) strains and 21 days after injection of isoproterenol (post-ISO) in 100 HMDP strains (a smaller number due to higher mortality of certain strains). Note that the significant correlation holds when considering separately pre-ISO (blue points, r=0.59p=21011) and post-ISO (red points, r=0.42p=1.5105) data. Lines show best fits of a linear model for pre-ISO (blue), post-ISO (red), and pre- and post-ISO combined (black). Expression data is taken from Santolini et al. (2018) and is averaged over all microarray probes for each gene.



Table 1
Patch clamp measurements of ICa,L, Ito,f, and IKur functional current density.

Mean current density averaged over n cells isolated from multiple hearts for each strain is given together with the standard error.

StrainICa,L (pA/pF)nIKur (pA/pF)nIto,f (pA/pF)nIKss (pA/pF)n
A/J13.71947 ± 1.230851911.61804 ± 2.790891313.03735 ± 1.64818145.23445 ± 0.4837713
BALB/cByJ10.72278 ± 1.3951911.496 ± 2.03274107.46938 ± 0.8261598.197 ± 0.6243810
BTBR T+tf/J14.75667 ± 1.14159616.42364 ± 2.78295119.271 ± 1.43985106.093 ± 0.395539
BXA12/PgnJ11.01333 ± 0.98995915.66429 ± 1.70548811.89875 ± 2.1802788.34556 ± 1.300659
BXA25/PgnJ17.57 ± 4.81376515.8049 ± 1.74509711.62704 ± 1.5961377.11986 ± 1.315546
BXH6/TyJ7.32625 ± 0.59327165.954 ± 0.43731108.73172 ± 0.67617117.98545 ± 0.8754511
C57BL/6J7.3925 ± 0.93181107.82727 ± 1.45134117.594 ± 0.92097109.16273 ± 1.6214811
CXB1/ByJ11.93909 ± 0.810221111.69556 ± 1.4209598.73883 ± 1.11087106.089 ± 0.572410
CXB11/HiAJ7.63625 ± 0.8934488.90316 ± 2.4975568.42537 ± 1.2463576.8481 ± 1.36387
AXB8/PgnJ7.355 ± 1.096126---
BXA14/PgnJ11.28091 ± 1.0079615---
BXA4/PgnJ11.77615 ± 1.1947613---
BXD34/TyJ11.12556 ± 1.137419---
CBA/J7.956 ± 0.8126910---
CXB7/ByJ9.05386 ± 0.608716---
SJL/J11.2745 ± 0.9970820---
Table 3
Heart mass before and 3 weeks after Isoproterenol (ISO) injection.
StrainHeart mass pre-ISO, mpre (g)Heart mass post-ISO, mpost (g)
BTBR T+tf/J0.141620.223
Table 2
Cell Shortening at 4 Hz pacing.
StrainΔL/L (4 Hz)n
BXA12/PgnJ0.0835 ± 0.01845
BXA14/PgnJ0.1182 ± 0.010814
BTBR T+tf/J0.0978 ± 0.00957
BALB/cByJ0.105 ± 0.01585
C57BL/6J0.0989 ± 0.0166
BXA25/PgnJ0.0838 ± 0.01125
Table 4
Reference values of ionic current parameters varied in the GES search.
ParameterDefinitionReference valueReference source


Ca2+ current flux333.32 mmol/(Cm C)Measured


Peak uptake rate1.17 μM/msChosen*


Peak NaCa rate36.6 μM/sBondarenko et al., 2004


Release current
12.9 sparks cm2/mAMahajan et al. (2008)


Ito,f peak
0.16 A/FMeasured


IKur peak
0.144 A/FMeasured
  1. *vup was chosen such that the reference Ca2+ transient amplitude was normal.

Table 5
Mouse ventricular myocyte model parameters.
Physical constants and ionic concentrations


Cell capacitance3.1 ×104μF


Cell volume2.58 ×105μl


Submembrane volume0.02 vi
FFaraday Constant96.485 C/mmol
RUniversal gas constant8.314 J mol1 K1
TTemperature298 K


External Na+
140 mM


Internal K+
143.5 mM


External K+
5.4 mM
[Ca2+]oExternal Ca2+
1.8 mM
Cytosolic buffering parameters


Troponin C
70 μmol/l cyt


on rate for Troponin C
0.0327 (μM ms)1


off rate for Troponin C
0.0196 (ms)1


SR binding site
47 μmol/l cyt


SR binding site disassociation constant0.6 μM


Calmodulin binding site concentration24 μmol/l cyt


Calmodulin binding site disassociation constant7 μM


Membrane binding site concentration15 μmol/l cyt


Membrane binding site disassociation constant0.3 μM


Sarcolemma binding
site concentration
42 μmol/l cyt


Sarcolemma binding
site disassociation constant
13 μM
SR release parameters


Spark lifetime10 ms


NSR-JSR diffusion time20 ms
 uRelease slope4 ms1


Release slope threshold90 μM / l cytosol


cp - cs diffusion time0.50 ms*


cs - ci diffusion time0.75 ms
Exchanger, uptake, and SR leak parameters


Uptake threshold0.5 μM


NaCa saturation


NaCa energy barrier


Ion mobility constant21 mM


Ion mobility constant87.5 mM


Ion mobility constant1380 μM


Leak current
1.74 × 105ms1
Ionic current parameters


Na+ current
13 mS/μF


Na+ background
current conductance
0.0026 mS/μF


Ca2+ background
current conductance
0.000367 mS/μ F


Strength of local LCC
calcium flux
9000 mM/(cm C)


IK1 conductance0.2938 mS/μF


INaK conductance1.716 mS/μF


IKss conductance0.025 mS/μF


Ito,s conductance0 mS/μF


Maximal JPMCA fluxone pA/pF


Saturation constant for Ca2+ current0.5 μM


Constant0.00054 cm/s


  1. *We have reduced this value from the original value of Mahajan et al. (2008) so that the Ca2+ transient increases when SERCA uptake rate is increased.

Table 6
Simulation outputs corresponding to physiological sensors
AbbreviationDescriptionReference value


Average cytostolic Ca2+ over one beat0.24 μM


Ca2+ transient amplitude0.5 μM


Diastolic Na+14 mM

Data availability

Gene expression data has been deposited in GEO under accession code GSE48760

The following previously published data sets were used
  1. 1
    NCBI Gene Expression Omnibus
    1. CD Rau
    2. J Wang
    3. Y Wang
    4. AJ Lusis
    ID GSE48760. Transcriptomes of the hybrid mouse diversity panel subjected to Isoproterenol challenge.

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