Variability and compensation of cardiomycoyte ionic conductances at the population level

Abstract

Conductances of ion channels and transporters controlling cardiac excitation may vary in a population of subjects with different cardiac gene expression patterns. However, the amount of variability and its origin are not quantitatively known. We propose a new computational method to predict this variability that consists of finding combinations of conductances generating a normal intracellular Ca²⁺ transient without any constraint on the action potential. Furthermore, we validate experimentally its predictions using the Hybrid Mouse Diversity Panel, a model system of genetically diverse mouse strains that allows us to quantify inter-subject versus intra-subject variability. The method predicts that conductances of inward Ca²⁺ and outward K⁺ currents compensate each other to generate a normal Ca²⁺ transient in good quantitative agreement with current measurements in ventricular myocytes from hearts of different isogenic strains. Our results suggest that a feedback mechanism sensing the aggregate Ca²⁺ transient of the heart suffices to regulate ionic conductances.

Introduction

Following the landmark publication of the Hodgkin-Huxley model of nerve-cell action potential over six decades ago Hodgkin and Huxley (1952), electrophysiological models of increasing complexity have been developed to describe the cardiac action potential (AP) and its interaction with the intracellular calcium (Ca²⁺) signal, which links electrical signaling to mechanical contraction in cardiomycoytes Bers (2001). As illustrated in Fig. 1 for a mouse ventricular mycoyte Bondarenko et al. (2004), those models typically involve a large set of interacting cellular components that includes various voltage-gated membrane ion channels and transporters, the Na⁺/Ca²⁺ exchanger, and Ca²⁺ handling proteins such as the ryanodine receptor (RyR) Ca²⁺ release channels, which open in response to Ca²⁺ entry into the cell via L-type Ca²⁺ channels, and the sarcoplasmic reticulum (SR) Ca²⁺ ATPase (SERCA), which uptakes Ca²⁺ back into the SR. Ca²⁺ release and uptake from the SR causes a transient rise in cytosolic Ca²⁺ concentration, the calcium transient (CaT), which activates myocyte contraction. Those cellular-scale models have been traditionally constructed by piecing together separate mathematical models describing molecular-scale components in the same species (guinea pig Luo and Rudy (1991), mouse Bondarenko et al. (2004), rabbit Shannon et al. (2004); Mahajan et al. (2008), dog Fox et al. (2002), etc), and by sometimes mixing models in different species. Experimental measurements of voltage-current relationships and other properties used to develop those models are typically averaged over several cells from one or a few hearts. While such prototypical “population-averaged” (and even “species-averaged”) models have proven useful to investigate basic mechanisms of cardiac arrhythmias on cellular to tissue scales Krogh-Madsen and Christini (2012); Karma (2013); Qu et al. (2014), they fall short of predicting how different individuals in a genetically diverse population respond to perturbations (such as physiological stressors, ion channel mutations, drug or gene therapies, etc) affecting one or several cellular components.

Figure 1.

Schematic representation of the sarcolemmal currents and intracellular Ca²⁺ cycling proteins of the mouse ventricular myocyte model.

In a neuroscience context, the limitation of population-averaged models has been highlighted by pioneering theoretical and experimental studies by Abbot, Marder, and co-workers demonstrating that ion channel conductances can exhibit a high degree of activity-dependent plasticity as well as variability between individuals LeMasson et al. (1993); Siegel et al. (1994); Liu et al. (1998); Golowasch et al. (1999); Prinz et al. (2004); Schulz et al. (2006); Marder and Goaillard (2006); Grashow et al. (2009); Marder (2011); O’Leary et al. (2013). Theoretical studies along this line first originated from an attempt to explain why neurons can maintain fixed electrical activity patterns despite a high rate of ion channel turnover. This question was addressed by treating ion channel conductances as dynamical variables in models of neuronal activity and by using the intracellular Ca²⁺ concentration as an activity-dependent feedback mechanism regulating their values LeMasson et al. (1993); Siegel et al. (1994); Liu et al. (1998), a mechanism supported by experiments Golowasch et al. (1999). Those model neurons displayed remarkable properties such as the ability to modify their conductances to maintain a given behavior when perturbed or to develop different properties in response to different patterns of presynaptic activity. Subsequently, a different type of computational study in which model parameters (conductances and synaptic strengths of a circuit model of the crustacean somato-gastric ganglion) were varied randomly, demonstrated that a similar bursting activity could be obtained with multiple parameter sets, i.e. multiple “good enough solutions” (GES) Prinz et al. (2004). This prediction agreed qualitatively with an experimental study demonstrating that neurons of the same circuits obtained from different crabs could have markedly different ion channel densities, corresponding to different gene expression, but yet different circuits could generate similar bursting activities Schulz et al. (2006).

A later experimental study further showed that different circuits, while operating similarly under controlled conditions, could respond differently to perturbations such as serotonin addition, which increases the bursting frequency at a population level, but lowers it in some individuals Grashow et al. (2009). Those findings shed light on why pharmacological treatments may work in some individuals but not others. In addition, they suggest that the existence of different good enough solutions provides an evolutionary advantage for the survival of a genetically diverse population by allowing different individuals to better adapt to different environmental challenges so as to survive and restore the population.

Those studies and related ones in the broader context of systems biology Daniels et al. (2008); Gutenkunst et al. (2007); Transtrum et al. (2015) and cardiac electrophysiology Sarkar and So-bie (2009, 2010); Weiss et al. (2012); Sarkar et al. (2012); Britton et al. (2013); Groenendaal et al. (2015); Muszkiewicz et al. (2016); Krogh-Madsen et al. (2016) have produced a paradigm shift from population-averaged models, with unique fine-tuned parameter sets, to populations of models characterized by multiple parameter sets. In this new paradigm, each set representing a different individual in a population can produce a similar behavior under controlled conditions, but a starkly different response to perturbations for some individuals. This paradigm shift, however, creates new theoretical and experimental challenges.

On the theoretical side, an open question is how to search for GES representing different individuals in a population. The results of this search, which has been carried out using various methods (e.g. random search Prinz et al. (2004), multivariate regression analysis Sarkar and Sobie (2009, 2010), or genetic algorithms Groenendaal et al. (2015)), depend critically on what outputs are selected to constrain model parameters. To date, parameter searches in a cardiac context have been “AP centric”, focusing primarily on features of the membrane voltage (V_m) signal. Sarkar and Sobie showed that very different combinations of ion conductances can produce almost identical cardiac AP waveforms, albeit different CaT amplitudes, and that adding additional constraints on the V_m and Ca²⁺ signals can further constrain model parameters Sarkar and Sobie (2009, 2010). Groenendaal et al. Groenendaal et al. (2015) constrained model parameters using V_m traces with variable AP waveforms recorded from guinea pig cardiomyocytes under randomly timed electrical stimuli, as opposed to a unique AP waveform recorded during periodic pacing. This search yielded parameter sets that are potentially better suited to describe more complex aperiodic forms of V_m dynamics relevant for arrhythmias. Britton et al. Britton et al. (2013) observed experimentally a signi1cant variability in AP waveform in rabbit Purkinje 1bers and searched for model parameter combinations that reproduce this waveform variability. They then used those parameter sets to predict different effects of pharmacological blockade of cardiac HERG (I_Kr current) potassium channel in different subjects Britton et al. (2013).

All those GES searches have relied in part or entirely on using measured AP featuresSarkar and Sobie (2009, 2010); Sarkar et al. (2012); Britton et al. (2013); Groenendaal et al. (2015); Muszkiewicz et al. (2016); Krogh-Madsen et al. (2016) to constrain model parameters. However, given that there is no known voltage sensing mechanisms regulating ion channel expression, it is unclear if natural biological variability can be predicted based on AP features. Here, we adopt a different “Ca²⁺ centric” view Weiss et al. (2012), which postulates as in a neuroscience context Golowasch et al. (1999); LeMasson et al. (1993); Siegel et al. (1994); Liu et al. (1998); O’Leary et al. (2013) that model parameters are predominantly constrained by feedback sensing of Ca²⁺, and potentially other ions (e.g. Na⁺) affecting ion channel regulation. This assumption is motivated by the fact that the CaT is directly linked to mechanical contraction and hence to arterial blood pressure in the organism, which is subject to strong feedback regulation by baroreceptors in the autonomic nervous system. Hence, multiple parameter combinations producing a normal CaT could potentially represent different GES in a genetically diverse population. In addition, unlike voltage, intracellular concentrations of Ca²⁺ and Na⁺ ions ([Ca]_i and [Na]_i, respectively) have a known interactive role in transcriptional regulation of cardiac ion channel proteins and their function Rosati and McKinnon (2004). For example, the Ca²⁺/calcineurin/NFAT pathway regulates L-type Ca²⁺ channel (LCC) expression Qi et al. (2008) and Na⁺ modulates cAMP-dependent regulation of ion channels in the heart Harvey et al. (1991) including phosphorylation of LCCs via cAMP-dependent protein kinase Balke and Wier (1992). To test this hypothesis, we perform a GES search in which parameters of a mouse ventricular myocyte model are only constrained by CaT features and [Na]_i. This search yields GES with different conductances of the L-type Ca²⁺ current (I_Ca,L) and K⁺ currents (I_to,f and I_Kur) and reveals that conductances are strongly correlated due to compensatory effects of those currents on the CaT.

On the experimental side, a major challenge is to test whether different GES produced by any given search method are representative of different individuals in a genetically diverse population. Performing this test generally requires distinguishing quantitatively the variability of conductances and electrophysiological phenotype observed in cells extracted from the same heart (intra-heart cell-to-cell variability) from the variability of the same quantities between different subjects (inter-subject variability). Making this distinction is made extremely difficult by the fact that AP features and conductances vary signi1cantly between cells extracted from same region of the heart Banyasz et al. (2011); Groenendaal et al. (2015) and that regional (e.g. ventricular base-to-apex and epicardium to endocardium) variations of ion channel expression are also present. The existence of large intra-heart cell-to-cell variability, and the practical limitation that only a finite number of cells can typically be extracted from a single heart for current measurements, raises the question of whether it is feasible to distinguish electrophysiological parameters between genetically distinct individuals.

To cope with this challenge, we use here the Hybrid Mouse Diversity Panel (HMDP) that is a collection of approximately 100 well-characterized inbred strains of mice that can be used to analyze the genetic and environmental factors underlying complex traits. Because inbred strains are isogenic and renewable, we are able to use multiple hearts from the same strain to obtain enough statistics to differentiate quantitatively between intra-heart and inter-subject variability in conductances of key currents (I_Ca,L, I_to,f and I_Kur) affecting the AP and CaT of mouse ventricular myocytes from different strains. The results show that, despite large cell-to-cell variability, some strains have clearly distinguishable mean conductances (i.e. conductances averaged over all cells for the same strain). Mean conductances can vary by up to two-and-a-half fold between strains. The results further show that, remarkably, variations of mean conductances for individual strains follow the same correlation (I_Ca,L current is large or small when the sum of I_to,f and I_Kur currents is large or small, respectively) predicted by our computational Ca²⁺ centric GES search. The central hypothesis that parameters are constrained predominantly by features of the CaT (as a surrogate for arterial blood pressure) is further validated experimentally by showing that strains with very different conductances have similar contractile activity. In addition, we use tissue scale simulations to show that compensation remains effective at an organ scale despite large cell-to-cell variability within an individual heart. Finally, we use our findings to interpret the results of recent studies of cardiac hypertrophy and heart failure induced by a stressor in the HMDP Ghazalpour et al. (2012); Rau et al. (2014, 2017); Santolini et al. (2018).

Results

Effects of individual conductances on the calcium transient

We first used a mouse ventricular myocyte model (see Methods) to investigate the effects of changing a single electrophysiological parameter on the CaT. The CaT was characterized by its amplitude, defined as the difference Δ[Ca]_i between the diastolic and peak value of the cytosolic Ca²⁺ concentration [Ca]_i, and the time-averaged value of [Ca]_i over one pacing period, denoted by ⟨[Ca]_i⟩. The CaT amplitude Δ[Ca]_i is a major determinant of the contractile force while ⟨[Ca]_i⟩ provides an average measure of the cytosolic Ca²⁺ concentration in the cell. Both quantities will be used as Ca²⁺ sensors for our multi-parameter search of GES and examining individual parameter effects will be useful later to interpret the results of that search. We vary the conductances of sarcolemmal currents and transporters depicted in Fig. 2A and the expression levels of Ca²⁺ handling proteins that include the ryanodine receptor (RyR) Ca²⁺ release channels and the sarcoplasmic reticulum (SR) Ca²⁺ ATPase SERCA, which pumps Ca²⁺ from the cytosol into the SR. For each parameter value, we pace the myocyte at a 4 Hz frequency for many beats until a steady-state is reached where the CaT pro1le used to calculate Δ[Ca]_i and ⟨[Ca]_i⟩ and the intracellular sodium concentration [Na] no longer vary from beat to beat.

Figure 2.

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/G_ref where G is the individual conductance value and G_ref some 1xed reference value. B) Time-averaged [Ca]_i over one pacing period (⟨[Ca]_i⟩) versus G/G_ref. Illustration of the effect of varying I_Ca,L conductance (C) and I_to,f conductance (D) on AP and CaT pro1les, where 50%, 100%, and 150% correspond to G/G_ref=0.5, 1.0, and 1.5, respectively. E) Effect of varying RyR conductance on SR Ca²⁺ concentration [Ca]_SR and CaT.

Fig. 2 shows the effects of individual parameter changes on the steady-state CaT amplitude (Fig. 2A) and average [Ca]_i (Fig. 2B). Both quantities are plotted as a function of conductance fold change G/G_ref where G_ref is a reference value producing a normal CaT. Increasing the conductance of the inward L-type Ca²⁺ current I_Ca,L is seen to strongly increase both [Ca]_i and ⟨[Ca]_i⟩ but has a weak effect on the AP waveform (Fig. 2C). The effects on the CaT stem from the fact that I_Ca,L is the main trigger of Ca²⁺-induced Ca²⁺ release (CICR), which transfers a large amount of Ca²⁺ from the SR to the cytosol. In contrast, increasing the conductances of K⁺ currents that are dominant in the mouse such as I_to,f (fast inactivating component of the transient outward current) and I_Kur causes both Δ[Ca]_i and ⟨[Ca]_i⟩ to decrease. Increasing either of those K⁺ currents speeds up repolarization (as illustrated for I_to,f in Fig. 2D) and hence inactivation of I_Ca,L, thereby reducing the magnitude of CICR. Increasing the conductance of the sodium-calcium exchanger current I_NaCa also causes both [Ca]_i and ⟨[Ca]_i⟩ to decrease by enhancing the forward mode of this current that extrudes Ca²⁺ from the cytosol. Changing RyR expression from its reference value is seen to leave [Ca]_i and ⟨[Ca]_i⟩ almost unchanged, even though it strongly affects the SR Ca²⁺ concentration [Ca]_SR (Fig. 2E). This behavior reflects the well-known effect that making RyR channels more leaky (e.g. by addition of caffeine that increases RyR activity or, similarly here, by increasing the magnitude of the Ca²⁺ release flux through RyRs) yields a transient increase in CaT amplitude, but no change in the steady-state CaT amplitude after [Ca]_SR adjusts to a lower steady-state level Bers (2001). This effect is illustrated by time traces of [Ca]_SR and [Ca]_i in steady-state for different RyR expression levels in Fig. 2E. Finally, changing the expression level of SERCA has opposite effects on Δ[Ca]_i and ⟨[Ca]⟩. Increasing SERCA magnitude increases SR Ca²⁺ load, thereby increasing the amount of SR Ca²⁺ elease and CaT amplitude, but at the same time depletes Ca²⁺ from the cytosol.

Computationally determined good enough solutions

Next, we performed a computational search for combinations of parameters that yield a normal electrophysiological output as defined by the steady-state CaT amplitude Δ[Ca]_i, time averaged cytosolic Ca²⁺ concentration ⟨[Ca]_i)⟩, and intracellular sodium concentration [Na]_i at a 4 Hz pacing frequency. A GES search that uses the diastolic and peak [Ca]_i values as Ca²⁺ sensors, instead of Δ[Ca]_i (the difference between the peak and diastolic [Ca]_i values) and time averaged [Ca]_i, gives nearly identical results. So our Ca²⁺ sensors can be straightforwardly interpreted physiologically as requirements of normal diastolic and systolic contractile function necessary for a normal arterial blood pressure at the organism scale. A “good enough solution” (GES) was defined as a combination of electrophysiological parameters that produces output values of those three quantities that are close enough to normal target values, which are defined as the values Δ[Ca]_i*,⟨[Ca]_i⟩*, and corresponding to the reference set of parameters (G_ref values) of the ventricular mycoyte model. The search was conducted by defining a cost function which is an aggregate measure of the deviation of output sensors s_n(p) from their desired target values . Here, N = 3 with s₁ = [Ca]_i, s₂ = [Ca]_i, and s₃ = [Na]_i, and ε is a small tolerance that we choose to be 5%. E is a function of model parameters p = (p₁, p₂, …) chosen to consist of the conductances of I_Ca,L, I_to,f, I_Kur, and I_NaCa as well as RyR and SERCA expression levels. Effects of individual changes of those parameters on CaT properties measured by s₁ and s₂ are shown in Figs. 2A,B. Conductances of other sarcolemmal currents that were found to have a negligible effect on the CaT were kept constant. The search for GES was conducted by first generating a large population of ∼ 10, 000 randomly chosen candidate models, with each model represented by a single parameter set p. A candidate model was generated by randomly assigning each parameter (p₁, p₂, …) a value comprised between 0 and 300% of its reference value G_ref. We then utilized a multivariate minimization algorithm (see Methods for details) that evolves p until the GES optimization constraint defined by Eq. 1 is satis1ed. This method typically yields a large number of GES (7263 of the ∼ 10, 000 trials yield a GES with 6 parameters and 3 sensors described above, with 2737 either not converging or not producing a physiological output) and is computationally more efficient than a random search without optimization that yields very few GES.

Results of the GES search are shown in Fig. 3. Fig. 3A shows the parameters of six representative GES and their corresponding AP waveforms (Fig. 3B) and CaT pro1les (Fig. 3C). The CaT pro1les are all very close to each other, which holds for all GES, while the AP waveforms exhibit larger variations owing to the fact that the GES search does not involve any voltage sensing. Fig. 3D shows histograms of parameters for all GES. Conductances of sarcolemmal currents tend to be highly variable except for I_NaCa, which turns out to be constrained by the intracellular sodium concentration sensor (s₃ = [Na]_i). This is revealed by a GES search with only Ca²⁺ sensing (s₁ and s₂) that yields a broader histogram for the I_NaCa conductance (Figure 3–Figure Supplement 1). The histogram of RyR expression level is very broad. This is consistent with the fact that this parameter was found to have a very weak effect on the CaT (Figs. 2A,B) due to the compensatory adjustment of SR Ca²⁺ load (Fig. 2E). In contrast, the histogram of SERCA is very narrow. This feature, which persists even if Na⁺ sensing is removed (Figure 3–Figure Supplement 1), is predominantly due to Ca²⁺ sensing. It stems from the fact that changing SERCA expression level has opposite effects on the CaT amplitude (Fig. 2A) and average [Ca]_i (Fig. 2B), increasing one while decreasing the other or vice-versa. Therefore, those opposite effects cannot be compensated by changes of conductance of sarcolemmal currents that simultaneously increase or decrease both Ca²⁺ sensors, or by changes of RyR expression level that has a negligible effect on the CaT due to SR load adjustment. However, conductances of inward and outward currents that change both Ca²⁺ sensors in opposite directions can in principle compensate each other. This compensation is revealed by representing each GES as a point in a 3D plot (Fig. 3E) whose axes are the conductances of I_Ca,L, I_to,f and I_Kur. This plot shows that all GES lie close to a 2D surface in this 3D conductance space due to a 3-way compensation between the effects of I_Ca,L, I_to,f, andI_Kur on the CaT. GES lie inside a smeared 2D surface (i.e. a 2D surface of finite thickness) in the 3D conductance space of Fig. 3E. This feature stems from the fact that the GES parameter space considered here is in principle 6-dimensional (4 sarcolemmal current conductances and 2 Ca²⁺ protein expression levels). However, SERCA expression and I_NaCa conductance are constrained by Ca²⁺ and Na⁺ sensing, respectively, and RyR expression has a negligible effect on both Ca²⁺ and Na⁺ sensors, thereby reducing the relevant parameter space to the 3 conductance axes of Fig. 3C. The subspace of GES that minimizes the cost function E must therefore lie on the 2D surface E = 0. This surface is smeared because I_NaCa is only constrained by Na⁺ sensing within a finite range and the GES search only minimizes E within a finite tolerance (E≤ϵ instead of E = 0).

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.

Figure 3.

Computationally determined good enough solutions (GES). 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 pro1les are shown in B) and C), respectively. D) Histograms of individual normalized conductances G/G_ref for a collection of 7,263 GES showing that some conductances are highly variable while others are highly constrained. E) Three-dimensional (3D) plot revealing a 3-way compensation between conductances of I_Ca,L, I_to,f, and I_Kur. 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 2-way compensation between pairs of conductances. F) Alternate representation of 3-way compensation obtained by plotting I_Ca,L versus the sum of I_to,f and I_Kur. 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.

To facilitate the comparison with experiments presented in the next subsection, it is useful to represent the 3-way compensation between I_Ca,L, I_to,f, and I_Kur conductances by plotting the sum of the peak currents of I_to,f and I_Kur versus the peak current of I_Ca,L with all three currents measured under voltage-clamp with a step from −50 to 0 mV. Those peak currents are proportional to conductances up to proportionality factors fixed by intra-and extra-cellular ionic concentrations and voltage. In this peak-current representation, the smeared 2D surface of GES of Fig. 3E takes on the simpler form of a thick nearly straight line (Fig. 3F).

Good enough solutions in the HMDP

In order to test the computational modeling predictions, and at the same time differentiate intra-heart cell-to-cell from inter-subject variability, we performed electrophysiological and contractile measurements on ventricular myocytes obtained from mouse hearts of 9 different strains from the HMDP listed in the Methods, each strain assumed to represent a different good enough solution. Peak values of I_Ca,L, I_to,f, and I_Kur were measured under voltage-clamp with a step from −50 to 0 mV following established protocols (see Methods). The K⁺ currents were measured in the same cell and the Ca²⁺ currents in different cells. Contraction was analyzed by measuring mycoyte shortening during several paced beats in separate cells for 6 strains that include 5 of the strains in which conductances were measured. In order to collect enough statistics to distinguish cell-to-cell from inter-strain variability, several hearts of each isogenic strain were used. The number of cells that could be obtained from one heart for L-type Ca²⁺ current, several K⁺ currents, or contraction analysis varied from 1 to 7 so that several hearts of each strain were needed to obtain enough independent measurements to statistically distinguish intra-heart cell-to-cell from inter-strain variability (see data in Methods).

The results of current and contraction measurements are shown in Fig. 4. In Fig. 4A we plot the sum of the peak currents of I_to,f and I_Kur versus the peak current of I_Ca,L together with the standard errors of the mean (SEM) of those quantities. Bar plots showing mean current values together with both SEM and standard deviation (SD) characterizing cell-to-cell variability are given in the Methods. We also superimpose on this plot the computationally predicted GES of Fig. 3F. Different HMDP strains, each representing a GES, are seen to function with different combinations of Ca²⁺ and K⁺ currents that compensate each other in a non-trivial 3-way fashion that closely follows the GES computationally determined with a 3-sensor search in which both the CaT and Na⁺ concentration are constrained (faded red points in Fig. 4A). The sum of I_to,f and I_Kur follows a linear correlation with I_Ca,L (p=0.0007) using 8 out of 9 strains and the correlation remains statistically signi1cant (p=0.0144) if the outlier strain (BXA12/PgnJ) is included. Interestingly, this outlier strain still falls within the larger range of computationally predicted GES using a 2-sensor search without Na⁺ sensing (faded blue points in Fig. 4A). To distinguish cell-to-cell from inter-strain variability, we performed a one-way ANOVA F-test on the I_Ca,L measurements. The result shows that I_Ca,L measurements for all strains do not originate from the same distribution (p-value p=0.000024). Furthermore, we performed a student T-test using raw data of I_Ca,L measurements for all pairs of strains. The results yield very small statistically signi1cant p-values for pairs of strains with sufficiently different average current values (e.g. BXA25/PgnJ, CXB1/ByJ, and C57BL/6J in Fig. 4A). Those results are consistent with the fact that mean currents differ much more than their standard error for those strains, as can be seen by visual inspection of means and SEM values corresponding to thin bars on both axes of Fig. 4A. We conclude that inter-strain variability of ion channel conductances can be distinguished from cell-to-cell variability of those same quantities for a signi1cant number of the strains investigated. While the Ca²⁺ and K⁺ currents were measured for the 9 strains reported in Fig. 4A the Ca²⁺ current was measured in 7 additional strains (total of 16 strains). Those additional measurements reported in the Methods con1rm that some strains can have markedly different I_Ca,L conductances.

Figure 4.

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 Fig. 3F showing the sum of I_to,f and I_Kur versus I_Ca,L for 9 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 1lled squares) are shown together with standard errors of the mean (thin bars) for each strain. Computationally determined GES are superimposed and shown as faded red points using all 3 sensors (CaT amplitude, average [Ca]_i, and diastolic [Na]_i) and faded blue points for 2 sensors (CaT amplitude and average [Ca]_i). Lines represent linear regression 1ts 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 1t validate the computationally predicted 3-way compensation of Ca²⁺ and K⁺ currents. The 3 strains selected for the organ scale study (C57BL/6J, CXB1/ByJ, and BXA25/PgnJ) with low, medium, and high I_Ca,L conductance, respectively, are highlighted by blue circles. B) Cell shortening 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 signi1cant differences in cell shortening between strains (p = 0.4136) supporting the hypothesis that different combinations of conductances produce a similar CaT and contractile activity.

Unlike ion channel conductances, CaT properties were assumed not to vary in the computationally-enabled GES search, which rests on the hypothesis that Ca²⁺ sensing provides a feedback mechanism that regulates ion channel gene and protein expression. The results in Fig. 4B which use contraction as a surrogate for CaT amplitude, support this hypothesis by showing that mean values of cell shortening do not vary substantially across strains. This is con1rmed by performing a standard ANOVA statistical test, which shows that cell shortening measurements for the six strains reported in Fig. 4B do not originate from different distributions within statistical uncertainty (p-value p=0.4136).

Compensation at the organ scale

Current measurements discussed in the previous section show that mean conductances of Ca²⁺ and K⁺ currents vary between strains in a compensatory way so as to produce a normal CaT. They also show that conductances vary signi1cantly from cell to cell around their mean values. This is illustrated in Fig. 5A for 3 mouse strains that have statistically distinguishable mean I_Ca,L conductances (low, medium, and high) as measured by standard errors (thick bars), but exhibit large cell-to-cell variability as measured by the standard deviations (thin bars) of the distributions of conductance measurements in individual cells. This raises the question of whether compensation remains operative at the organ scale in the presence of large cell-to-cell variability. There are two interlinked aspects to this question. The first relates to the cellular-level dynamical coupling between membrane voltage and intracellular Ca²⁺ dynamics that is inherently nonlinear Krogh-Madsen and Christini (2012); Karma (2013); Qu et al. (2014). Even when cells are uncoupled, this nonlinearity could potentially cause the mean CaT amplitude in an ensemble of cells with highly variable conductances to differ from the CaT amplitude computed in a single cell with conductances set to the mean values of the ensemble, as traditionally done in cardiac modeling. The second aspect relates to the additional effect of gap-junctional coupling between cells. This effect is well-known to smooth out cell-to-cell variation of AP waveforms on a mm scale that is much larger than the individual mycoyte length. However, whether this smoothing translates into increased organ-scale uniformity of CaT amplitude and contractility is unclear.

Figure 5.

Organ scale compensation. A) Mean I_Ca,L conductance in 3 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 3 strains by assigning normally distributed random values to the I_Ca,L, I_to,f and I_Kur conductances using experimentally determined means and standard deviations. The blue, green, and red points correspond to the 3 HMDP strains with low (C57BL/6J), medium (CXB1/ByJ), and high (BXA25/PgnJ) I_Ca,L conductance, respectively, and the grey points are the results of the 3-sensor GES search (same as Fig. 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 signi1cantly 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 (Ca) 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 3 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 I_Ca,L conductance and with I_to,f and I_Kur conductances 1xed to their reference values. Lack of compensation between Ca²⁺ and K⁺ currents in this case yields different mean CaT amplitude.

Figure 3–Figure supplement 2.

Correlation between I_Ca,L and the sum of I_to,f and I_Kur is weaker but still signi1cant when intracellular sodium concentration is not constrained.

Figure 4–Figure supplement 4.

Patch clamp measurements of mean I_Ca,L (A), I_to,f (B), and I_Kur (C) functional current density averaged over multiple cells for 9 HMDP mouse strains with standard errors (thick bars) and standard deviations (thin bars).

Figure 5–Figure supplement 5.

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.

To address those two aspects, we constructed tissue scale computational models for 3 mouse strains with statistically distinguishable average conductances (as illustrated for I_Ca,L in Fig. 5A). Tissues of each strain consisted of 56 × 56 electrically coupled cells (see Methods for details). Simulations were carried out with and without electrical coupling to asses the effect of the latter. The conductances of I_Ca,L, I_to,f, and I_Kur were assumed to vary randomly from cell to cell. Their values were drawn randomly from Gaussian distributions with average values and standard deviations that match experimental current measurements in each strain. All other parameters were kept fixed to reference values. The resulting cell-to-cell variation of conductances is shown in Fig. 5B) using the same peak-current representation of Figs. 3F and 4A. In this representation, each point represents a different cell, and clouds of points of the same color represent all cells in a tissue of the same strain. Furthermore, the center of each cloud falls on the thick line corresponding to the computationally determined GES surface where compensation is operative at the single cell level.

The results of simulations with populations of uncoupled and coupled cells with randomly varying conductances are shown in Figs. 5C to 5G. Fig. 5C shows that AP waveforms are highly variable when cells are uncoupled, reflecting the variability in conductances. Fig. 5D shows that AP waveforms becomes uniform when cells are coupled, as expected, even though interstrain variability is still signi1cant. Fig. 5E compares histograms of CaT amplitude and AP duration (APD) when cells are uncoupled and coupled. Consistent with the AP waveforms of Figs. 5C and 5D, APD histograms in Fig. 5E show that junctional coupling strongly reduces APD variability, as expected. CaT amplitude histograms in turn reveal that, in coupled cells, the more uniform APD translates into a more uniform CaT amplitude (i.e. a narrower Ca histogram), reflecting the strong effect of the cell’s APD on its CaT amplitude. At the organ scale, this ensures that cells in tissue contract with similar force (proportionate to the CaT amplitude), so that force generation is homogeneous throughout the tissue.

Fig. 5F and 5G show that compensation remain operative at the organ scale. Fig. 5F shows that, even though the strains have different average conductances (Fig. 5B), they produce CaT amplitude histograms with approximately the same mean and width. In contrast, if the same simulation is repeated by 1xing the conductance of K⁺ currents to the value of the strain with the medium value of I_Ca,L conductance (CXB1/ByJ), the different I_Ca,L conductances are not compensated by different I_to,f and I_Kur conductances, yielding CaT amplitude distributions with shifted peaks and hence different aggregate contraction (Fig. 5G).

In summary, our results show that compensation remains operative at the organ scale because CaT amplitude histograms have similar means with and without electrical coupling (Fig. 5E). This implies that cell-to-cell variability of conductances and hence APD causes variability of CaT amplitude without signi1cantly affecting its mean, so that I_Ca,L and potassium currents can compensate each other even though conductances exhibit large cell-to-cell variations from their mean values. Gap junctional coupling has the additional important effect of reducing CaT amplitude variability, thereby promoting tight organ-level behavior despite high cell-to-cell variability.

Cardiac hypertrophic response to a stressor

From a functional standpoint, the most relevant implication of the present study is that different GES may exhibit markedly different responses to perturbations, as previously demonstrated in a neuroscience context Grashow et al. (2009). To examine this possibility, we reviewed data from separate studies of isoproterenol (ISO)-induced cardiac hypertrophy and heart failure in approximately 100 HMDP strains that include most of the strains used in the present study. In those studies, heart mass was measured in those strains before (m_pre) and three weeks after (m_post) implantation of a pump continuously delivering isoproterenol (Table 3). Fig. 6 reveals the existence of a statistically very signi1cant correlation between baseline I_Ca,L conductance and the hypertrophic response (m_post/m_pre). Although many factors contribute to the hypertrophic response in the HMDP Rau et al. (2017); Santolini et al. (2018), intracellular Ca²⁺ overload activating the Ca²⁺-calcineurin-NFAT signaling pathway has been shown to play a major role Bers (2008). Since I_Ca,L is the major pathway of Ca²⁺ entry into the cytoplasm, it is intriguing to speculate that strains with a larger baseline I_Ca,L conductance under baseline conditions have a more robust increase in I_Ca,L that is not adequately compensated by repolarizing K⁺ currents, making those strains more susceptible to Ca²⁺ overload when I_Ca,L is enhanced during sustained β-adrenergic stimulation by isoproterenol. Hypothetically, this may result in a stronger cardiac hypertrophic response. To make this case convincingly, how-ever, would require demonstrating that Ca²⁺ overload is chronically worsened in strains with a high baseline I_Ca,L conductance and ruling out other strain-dependent hypertrophy-promoting pathways that are not Ca²⁺-dependent, which is beyond the scope of the present work.

Figure 6.

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

In a neuroscience context, ionic conductances of neurons from the stomatogastric ganglion of different crabs were previously found by Schulz et al. (2006) to be correlated with gene expression, as shown by independent measurements in the same subjects of functional densities of different ion channels, used to determine conductances, and mRNA levels of genes encoding for pore forming subunits of those channels. In the present study, we did not perform independent measurements of gene expression in the same ventricular myocytes used to measure ionic conductances. However, to examine the possible relationship between compensation of conductances and gene expression, we reviewed the gene expression data from the aforementioned studies of ISO-induced cardiac hypertrophy and heart failure in approximately 100 HMDP strains that include most of the strains used in the present study. Gene expression was measured both in control (pre-ISO) and after injection of ISO for 21 days in 8-10 week old female mice (post-ISO). Details of heart biopsies conducted pre-and post-ISO and microarray data analysis are given in the Methods section of Santolini et al. (2018).

No statistically signi1cant pairwise correlation (Pearson correlation coefficient r < 0.25 and p-value p > 0.05) were found between expression levels of the genes Cacna1c, Kcnd2, and Kcna5, encoding for the pore forming subunit of the Cav1.2, Kv4.2, and Kv1.5 channels associated with I_Ca,L, I_to,f, and I_Kur, respectively. However, a statistically very signi1cant correlation was found between expression levels of Cacna1c and Kcnip2 that encodes the KChIP2 accessory β subunits directly interacting with Kv4.2 (see Fig. 7 and caption for r and p values). This correlation is present both in control (pre-ISO), which is the condition relevant to our conductance measurements in selected strains (Fig. 4), and post-ISO. Since increased KChiP2 level is known to increase the functional current density of I_to,f Kuo et al. (2001); Jin et al. (2010), the strong positive correlation between Cacna1c (Cav1.2) and Kcnip2 (KChIP2) expression levels may partially contribute to the positive correlation between I_Ca,L and I_to,f+I_Kur functional current densities (Fig. 4).

Figure 7.

Compensation and gene expression. Plot showing the existence of a statistically very signi1cant correlation (Pearson correlation coefficient r = 0.47 and p-value p = 8.1 10^-13) between the expression level of Kcnip2, encoding the KChIP2 accessory β subunits that interact with Kv4.2 channels (I_to,f) and of Cacna1c, a gene encoding the a1C subunit of the Cav1.2 L-type calcium channels (I_Ca,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 signi1cant correlation holds when considering separately pre-ISO (blue points, r = 0.59, p = 2 10^-11) and post-ISO (red points, r = 0.42, p = 1.5 10^-5) data. Lines show best 1ts 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.

Discussion

In the present study, we have proposed a new methodology for searching for combinations of electrophysiological parameters representing different individuals in a genetically diverse population. While previous studies have used primarily AP features to constrain parameters Sarkar and Sobie (2009, 2010); Weiss et al. (2012); Sarkar et al. (2012); Britton et al. (2013); Groenendaal et al. (2015); Muszkiewicz et al. (2016); Krogh-Madsen et al. (2016), we have chosen to constrain parameters using the Ca²⁺ transient that plays a key role to regulate ion channel expression and activity. We have also examined the effect of additionally constraining the intracellular Na⁺ concentration that is also known to modulate ion channel activity. Regulatory mechanisms traverse different levels of biological organization from transcriptional regulation to post-transcriptional and post-translational modi1cation to ion channel trafficking and phosphorylation. Those mechanisms are presently not known in sufficient detail to be modeled quantitatively. However, there is sufficient experimental evidence of feedback sensing of cellular activity via Ca²⁺ and Na⁺ concentrations to make a search that constrains model parameters based on those signals plausible. The Ca²⁺ transient determines the contractile force underlying arterial blood pressure generation regulated by baroreceptor feedback via the autonomic nervous system. Hence, 1xing the diastolic and peak [Ca]_i values is a physiologically meaningful choice to search for parameter combinations that produce a normal diastolic and systolic function, which we have adopted here.

A remarkable and nontrivial finding of the present computational study is that this choice suffices to produce a physiological AP waveform whose duration matches the experimentally observed variation in isolated myocytes even though the voltage signal is not used to constrain model parameters. This is further highlighted by the fact that, even though the GES search is performed using target values of Ca²⁺ sensors for a 4 Hz pacing frequency, different GES exhibit similar force-frequency relationships over a broad range of pacing frequencies from 0.25 to 4 Hz. Force-frequency curves (with CaT amplitude used as force equivalent) for parameters corresponding to three different strains with low, medium and high L-type Ca²⁺ current conductance all exhibit a negative staircase (i.e. increased CaT amplitude with decreasing frequency < 1 Hz) characteristic of mouse electrophysiology Bers (2001). This property stems from the knock on effect of voltage on the L-type Ca²⁺ current and SR Ca²⁺ release via CICR. As a result of this effect, constraining the CaT indirectly constrains the relative magnitudes of depolarizing and repolarizing currents affecting the AP, i.e. the same CaT amplitude can be obtained with combinations of Ca²⁺ and K⁺ currents that are both large or both small, thereby compensating each other, but not with combinations in which the Ca²⁺ current is large and the sum of K⁺ currents is small or vice-versa. It remains that the AP waveform and duration are only partially constrained by the CaT and are thus more variable than in a GES search that uses AP features such as duration, plateau voltage, etc, to constrain parameter sets Sarkar and Sobie (2009, 2010).

While the additional constraint to keep the intracellular Na⁺ concentration [Na]_i within a normal physiological range is not necessary to produce a physiological AP waveform, it constrains more tightly the conductance of the Na⁺-Ca²⁺ exchanger current (compare I_NaCa histograms of Fig. 3D and Figure 3–Figure Supplement 1). This leads in turn to a tighter 3-way compensation between the L-type Ca²⁺ current and two dominant repolarizing K⁺ currents in the mouse, which is less pronounced in a GES search in which [Na]_i is not constrained (Figure 3–Figure Supplement 2) and I_Ca,L can be compensated by I_NaCa in addition to those K⁺ currents (4-way compensation).

The present GES search can identify possible combinations of electrophysiological parameters underlying a normal cardiac electrophysiological phenotype. However, it cannot by itself predict which GES among those found are actually represented, even approximately, in a genetically diverse population. Previous electrophysiological studies have indicated that ion channel/transporter conductances and AP waveform can vary from cell to cell and between different subjects. However, distinguishing intra-heart cell-to-cell variability from inter-subject variability has so far proven difficult since different subjects have different genetic backgrounds. Therefore, electrophysiological experiments that use cells isolated from the heart of a subject with a speci1c genetic background cannot be assumed to be repeatable in other subjects.

In the present study, we have leveraged the fact that different mice in the same HMDP inbred strain are isogenic to distinguish for the first time intra-heart cell-to-cell variability from intersubject (inter-strain in the HMPD context) variability. This has allowed us to use several hearts for each strain and perform current measurements in enough cells to statistically distinguish mean conductances of several currents in different strains. The results (Fig. 4A Table 1 and Figure 4–Figure Supplement 1) clearly show that conductances differ between strains. Statistical testing shows that I_Ca,L conductance measurements for different strains are very unlikely to belong to the same distribution (as indicated by the very small p-value) and mean conductances can vary by as much as two-hand-a-half fold between pairs of strains (e.g. C57BL/6J and BXA25/PgnJ), far in excess of the typical standard error of the mean. Furthermore, guided by the predictions of our computational GES search, we have also measured conductances of two dominant repolarizing currents in the mouse, I_to,f and I_Kur, to test for the existence of compensation between those currents and I_Ca,L. The results (Fig. 4A) show that for 8 out of the 9 strains in which all three currents were measured, the currents accurately compensate each other as predicted by the GES search in which both the CaT and [Na]_i are constrained. Compensation is evidenced by the linear regression 1t of I_to,f+I_Kur versus I_Ca,L. One outlier strain (BXA12/PgnJ) deviates from this 1t but still falls within the larger ensemble of computationally predicted GES without the [Na]_i constraint. Importantly, cells isolated from mouse strains with very different I_Ca,L conductance have statistically indistinguishable contractile function (Fig. 4B). This suggests that compensation between I_Ca,L and K⁺ currents in different HMDP strains is present to maintain Ca²⁺ homeostasis, as assumed in the computational GES search. As a whole, the results clearly support the hypothesis that Ca²⁺ concentration play a major role in feedback sensing of cellular activity and regulation of ion channel expression. While more strains would need to be studied to more accurately determine the role of the Na⁺ concentration, measurements reported in Fig. 4A suggest that it plays at least an auxiliary role in further constraining conductances beyond Ca²⁺ sensing.

Table 1.

Patch clamp measurements of I_Ca,L, I_to,f, and I_Kur functional current density. Mean current density averaged over n cells isolated from multiple hearts for each strain is given together with the standard error.

Finally, the correlation between I_Ca,L conductance and cardiac hypertrophic response of HMDP strains to sustained β-adrenergic stimulation (Fig. 6) also highlights the importance of considering the inherent variability of electrophysiological parameters in a genetically diverse population to interpret the variability of phenotypic response in the setting of complex diseases such as heart failure. It also suggests that pharmacological treatments such as anti-arrhythmic L-type calcium channel blockers, or pathologies such as hyperkalemia (elevated potassium level), will have different effects in different subjects. For example, in the setting of the HMDP, an L-type calcium channel blocker or hyperkalemia would be expected to have a stronger effect on the calcium transient and action potential of mice strains that function under normal conditions with larger I_Ca,L and potassium current conductances. Taking into account this population-variability seems ultimately needed to develop personalized therapies for cardiac arrhythmias and heart failure.

Methods and Materials

Overview of the HMDP

The hybrid mouse diversity panel (HMDP) consists of a population of over 100 inbred mouse strains selected for usage in systematic genetic analyses of complex traits Ghazalpour et al. (2012). The main goals in selecting the strains were to (i) increase resolution of genetic mapping, (ii) have a renewable resource that is available to all investigators world-wide, and (iii) provide a shared data repository¹ that would allow the integration of data across multiple scales, including genomic, transcriptomic, metabolomic, proteomic, and clinical phenotypes.

Electrophysiological and contraction measurements

Cell Isolation

Ventricular myocytes were enzymatically isolated from the hearts of adult female mice (8-12 wk old) using a procedure previously developed and utilized to isolate rabbit cardiomyocytes by Yang et al. Yang et al. (2008). Brie2y, hearts were removed from mice anesthetized with intravenous pentobarbital and perfused retrogradely at 37°C in Langendorff fashion with nominally Ca²⁺-free Tyrode’s buffer containing 1.2 mg/ml collagenase type II (catalog number 4176; Worthington) and 0.12 mg/ml protease type XIV (catalog number P5147; Sigma) for 10-17 min. After washing out the enzyme solution, the ventricles were cut from the atria and aorta and transferred to a separate glass dish containing Tyrode’s solution. Cells were isolated by gentle mechanical dissociation, stored at room temperature, and used within 5 h. This procedure typically yielded 30-50% of rod-shaped and Ca²⁺-tolerant myocytes.

Patch Clamping

Isolated ventricular myocytes were patch clamped in the whole cell ruptured patch con1guration using borosilicate glass pipettes (1-3-megaohm tip resistance). Myocytes were superfused at 34-36°C with Tyrode’s solution modi1ed accordingly. Currents were measured under voltage clamp conditions, using an Axopach 200B ampli1er with a Digidata 1440A interface (Axon Instruments, Union City, CA). Data were acquired and analyzed using pClamp (Axon instruments) and Origin (Origin).

L-type calcium current measurements

For characterization of I_Ca properties, the pipette solution, designed to eliminate K⁺ and Cl^- currents, contained (in mM) 100 CsMeS, 30 CsCl, 5 MgATP, 5 phophocreatine di(tris), 5 N-2-hydroxyethylpi-perazine-N’-2-ethanesulfonic acid (HEPES), 5 NaCl and 0.1 1,2-Bis(2-Aminophenoxy)ethane-N,N,N’,N’-tetraacetic acid (BAPTA) (pH adjusted with HEPES to 7.1-7.2). The superfusate, designed to eliminate K⁺ currents, contained (in mM), 136 NaCl, 5.4 CsCl, 1 MgC12, 0.33 NaH2PO4, 10 glucose, 5 HEPES, and 1.2 CaC12 (pH adjusted with Trizma base to 7.4). Voltage-clamp protocols to assess activation were as previously described by Delbridge et al. Delbridge et al. (1997). In all experiments, peak I_Ca current was recorded after a voltage step from −50 to 0 mV.

Potassium current measurements

For characterization of K⁺ current properties, the pipette solution, designed to eliminate Ca²⁺ currents, contained (in mM) 130 KCl, 5 MgATP, 5 phophocreatine di(tris), 5 HEPES, 5 NaCl, and 10 BAPTA (pH adjusted with HEPES to 7.1-7.2). The superfusate, designed to eliminate Ca²⁺ currents, contained (in mM) 136 NaCl, 5.4 KCl, 1 MgC12, 0.33 NaH2PO4, 10 glucose, 5 HEPES, and 0.2 CdCl2 (pH adjusted with Trizma base to 7.4). To access the activation of K⁺ current components: I_Kur, I_to,f and I_Kss, we adopted a voltage protocol, similar to the one reported by Zhou et al. (1998), in combination with the usage of 4-aminopyridine (4-AP) that has the following pharmacological properties: (1) I_Kur is markedly blocked by 4-AP at submillimolar concentration (e.g., 0.1 mM); (2) a higher concentration (i.e., > 1 mM) blocks I_to,f effectively; and (3) I_Kss is 4-AP resistant. Using this procedure, peak I_Kur, I_to,f and I_Kss currents could be deduced from three current measurements without 4-AP and with 0.1 mM and 2 mM 4-AP after a voltage step from −50 to 0 mV.

Results of patch clamp measurements

Results of patch clamp measurements for all strains are summarized in Table 1 and Figure 4–Figure Supplement 1. Both the L-type Ca²⁺ current and two K⁺ currents (I_to,f and I_Kur) were measured in 9 strains and the L-type Ca²⁺ current alone was measured in 16 strains.

Contraction analysis

The length of ventricular myocytes was measured and analyzed using the method described by Sdek et al. (2011). Brie2y, myocytes were imaged during pacing using a high-speed charge-coupled device-based camera (128 × 128 pixels; Cascade 128+; Photometrics) at 290 frames per second. The acquired video image data were then processed using Imaging Workbench software (version 6.0; INDEC BioSystems). Myocyte length was measured and analyzed using ImageJ software. Cell shortening was calculated from the ratio of peak systolic length to resting diastolic length averaged over 10 contractions evoked by the stimulus train. Results of cell shortening measurements for six strains are summarized in Table 2.

Table 2.

Cell Shortening at 1 Hz pacing

Table 3.

Heart mass before and 3 weeks after Isoproterenol (ISO) injection.

Heart extraction and mass measurement for cardiac hypertrophic response

At sacri1ce, hearts were excised, drained of excess blood and weighed. Each chamber of the heart (LV with inter-ventricular septum, RV free wall, RA and LA) was isolated and subsequently weighed. Cardiac hypertrophy was calculated as the increase in total heart weight after isoproterenol (ISO) treatment compared to control animals (see Table 3). As described prevoiously Wang et al. (2016), the ISO treatment consisted of 30 mg per kg body weight per day of Isoproterenol (ISO) administered for 21 days in 8-10 week old female mice using ALZET osmotic mini-pumps, which were surgically implanted intraperitoneally. The average number of control hearts per strain was 2.75. The average number of treated hearts per strain was 3.5.

Mathematical Model of Mouse Ventricular Myocytes

We have developed a novel mathematical model of mouse ventricular myocytes that combines elements of previously published ventricular mycoyte models Shiferaw et al. (2003); Shannon et al. (2004); Bondarenko et al. (2004); Mahajan et al. (2008). For this purpose, we kept the mathematical formulation of intracellular calcium cycling of the Mahajan model developed by Shiferaw et al. (2003) and replaced several sarcolemmal currents by those formulated by Bondarenko et al. (2004) for mouse ventricular mycoytes. This model allowed us to explore efficiently the space of good enough solutions that we can compare to experimentally measured variability found in the hybrid mouse diversity panel (HMDP). The Bondarenko et al. (2004) mouse model has sarcolemmal currents 1tted to detailed experimental measurements of sarcolemmal currents in mouse ventricular myocytes. The Mahajan et al. (2008) model is a rabbit model that integrates a Markov model of I_Ca,L together with the Shannon et al. (2004) formulation of other sarcolemmal currents and the Shiferaw et al. (2003) model of calcium cycling and SR calcium release. The Shiferaw et al. (2003) model represents the release of calcium from the SR as a sum of individual spark events, which reproduces important observed instabilities such as Ca²⁺ transient alternans. Even though the Shiferaw et al. (2003) model of calcium cycling model was developed for rabbit mycoytes, it can in principle describe Ca²⁺ cycling in other species including mouse. Therefore, we constructed a mouse model starting with the Mahajan et al. (2008) model, which incorporates the Shiferaw et al. (2003) model of calcium cycling, and using the formulation of Shannon et al. (2004) for I_Ca,L and Bondarenko et al. (2004) for other sarcolemmal currents 1tted to mouse data. We did not use the Bondarenko et al. (2004) detailed Markov formulations of the gating of I_Na and I_Ca,L channels with fast transition rates that were computationally prohibitive for the number of simulations we performed in this study. However, we checked that our combined model model reproduces well the mouse electrophysiological phenotype of the Bondarenko et al. (2004) model while being computationally efficient and incorporating a realistic description of Ca²⁺ cycling. We veri1ed that the Shiferaw et al. (2003) model of Ca²⁺ cycling integrated with the Shannon et al. (2004) model of I_Ca,L produced a normal bell-shaped SR release as a function of step-voltage. We also veri1ed that the force-frequency relationship produced by the model has the correct negative staircase observed experimentally. The equations for the model are described below. The reference values of the 6 parameters varied in this study are given in Table 4. Table 5 lists all other parameters used that were kept fixed.

Table 4.

Reference values of ionic current parameters varied in the GES search

Table 5.

Mouse ventricular myocyte model parameters.

Equations for Ca2+ cycling

We use the model for Ca²⁺ cycling developed by Shiferaw et al. (2003) and subsequently implemented in Mahajan et al. (2008). The equations for Ca²⁺ cycling are where the SR leak flux and RyR release flux are given by

Intracellular Ca2+ buffering

Similarly to Mahajan et al. (2008). All buffering parameters are experimentally based and summarized in Shannon et al. (2004). Buffering to SR, calmodulin, membrane, and sarcolemma binding sites are modeled using the instantaneous buffering approximation given by

Buffering to Troponin C is given by

The SERCA uptake pump

Similarly to Shiferaw et al. (2003)

Na+ dynamics

Intracellular Na⁺ dynamics are given by

In order to reduce computation time, we have sped up the rate at which the system reaches steady-state by increasing d[Na⁺]_i/dt by a factor of 100. This will make sodium converge to steadystate on a time-scale fast enough to perform the number of simulations necessary for this study. Once the cell reaches steady-state, d[Na⁺]_i/dt is zero, so this modi1cation will not affect the sodium dynamics at steady-state. By doing this, we can save up to 90% of calculation before the system reaches steady-state.

Ionic currents

The rate of change of the membrane voltage V is described by the equation where I_stim is the external stimulus current driving the cell.

The L-type Ca current (I_Ca,L)

Similarly to the Shannon et al. (2004) model,

Calcium background leak (I_Ca,b)

The Sarcolemmal Ca2+ ATPase (I_PMCA)

The sarcolemmal Ca²⁺ pump (I_PMCA) provides another mechanism, in addition to the exchanger (I_NaCa), for the extrusion of Ca²⁺ ions out of the cell. This pump is not included in Mahajan et al. (2008). We added this current using the formula used by Bondarenko et al. (2004).

The Na+-Ca2+ exchange flux (NaCa)

Similarly to the Bondarenko et al. (2004) model,

The fast sodium current (I_Na)

Similarly to the Shannon et al. (2004) model,

Sodium background leak (I_Na,b)

Inward recti1er K+ current (I_K1)

Similarly to the Bondarenko et al. (2004) model,

The fast component of the transient outward K+ current (I_to,f)

This current is modi1ed from the formulation of Bondarenko et al. (2004) as: where r = 0.37 accounts for the presence of a persistent outward potassium current in patch clamp measurements of I_t0,f. We have increased the rates of the inactivation gate (α _i and β_i) from the original formulation to match experimental measurements of I_to,f inactivation rate under voltage clamp.

The slow component of the transient outward K+ current (I_to,s)

Similarly to the Bondarenko et al. (2004) model,

The ultra-rapidly activating component of the delayed recti1er K+ current (I_Kur/I_K,slow)

Similarly to the Bondarenko et al. (2004) model,

We have reduced the timescale of inactivation (τ _iur) from the original formulation to match experimental measurements of I_Kur inactivation rate under voltage clamp.

The Non-inactivating steady-state K+ current (I_Kss)

Similarly to the Bondarenko et al. (2004) model,

The Na+-K+ pump current (I_NaK)

Similarly to the Bondarenko et al. (2004) model,

Effect of c_s-c_i diffusion rate on how the Ca²⁺ transient depends on SERCA

The effect of modi1cations of the SERCA pump on the steady-state Ca²⁺ transient is shown in Fig.1. While increasing the SERCA’s peak uptake current has the effect of sequestering Ca²⁺ back into the SR which would reduce the Ca²⁺ transient, the dominant effect is to increase the Ca²⁺ transient due to higher SR Ca²⁺ load at steady-state. This is consistent with experiments showing restoration of Ca²⁺ transient amplitude when SERCA is upregulated (del Monte et al.,; Del Monte et al., 2002).

Initial simulations using the original Ca²⁺ cycling parameters of Mahajan et al. (2008) found the SR Ca²⁺ load to decrease as the uptake rate was increased, as more Ca²⁺ was extruded from the submembrane region of the myocyte via NCX before it had an oppertunity to diffuse into the c_i compartment to be re-uptaken in into the SR. In order to rectify this, we increased the diffusion rate between the c_s and c_i compartments. We found that when this rate is faster (smaller τ _s) the Ca²⁺ increases with increasing SERCA uptake rate. For this reason we use a value of τ _s = 0.75 ms for all simulations in this study.

Computational reproduction of patch clamp experiments

In order to compare the GESs found by the computational search to the phenotype variability found in the HMDP, we iterate the model with voltage held constant, reproducing the experimental patch clamp procedure described in the Methods section of the main text. Each model is simulated for 1 s with V_m held at −50 mV in order to reach steady-state. V_m is then raised to 0 mV, and peak values I_Ca,L, I_to,f and I_Kur are recorded.

Tissue scale modeling

Tissue scale modeling is performed using a 56 × 56 array of myocytes, each with individual values of ionic conductances. Electrotonic coupling is simulated by introducing a diffusive into the V_m evolution equation,

The applied stimulus current occurs at a pacing rate of 4 Hz and is applied to each myocyte simultaneously. The diffusive term is applied isotropically, with diffusive co-efficient, D = 1 cm²/s. In the discretized diffusion equation,

We use a lattice size of Δx = 225µm, such that the 56 × 56 lattice represents a 1.25 cm × 1.25 cm tissue.

GES search

In this study we consider variation in 6 important ionic currents: L-type Ca current (I_Ca,L), the SR ATPase SERCA, Na⁺ - Ca²⁺ exchange (NaCa), ryanodine receptor (RyR), the transient outward K⁺ current (I_t0,f), and the ultra-rapidly-activating K⁺ current (I_Kur). The strength of these ionic currents is determined by their conductance g_i. Any given set of parameters (p = {p₁, p₂,., p_n}) corresponds to a different candidate myocyte model and produces a different phenotype, which we characterize by quanti1able measurements of its steady-state behaviour (sensors) i.e. steady state calcium transient amplitude, action potential duration and sarcoplasmic reticulum (SR) Ca²⁺ concentration. When stimulating a model with a given period, these parameters (once the simulation has reached steady state) produce a phenotype which we can compare to the phenotype produced by the standard parameters of our model (p_ref).

We define a cost function E that quanti1es how much each model’s phenotype differs from our reference phenotype as where the s_n(p)s are sensors characterizing the electrophysiological phenotype of the model’s output. E(p_ref) is zero by definition. The three sensors used in this study are listed in table 6.

Table 6.

Simulation outputs corresponding to physiological sensors

We define a good enough solution (GES) as a set of conductances with a phenotype such that the value of cost function E(p) is less than a threshold ∊ = 0.05. None of the cost function sensors s_n are based on the membrane potential, and therefore a GES does not necessarily have an action potential shape close to the reference action potential shape. A GES is required to achieve steadystate. Therefore parameters that produce parameter sets that do not reach a steady state during pacing at constant cycle length, such as those which exhibit calcium transient alternans, are not considered GESs. We additionally reject any set of parameters for which the output steady-state SR Ca²⁺ load is above a threshold [Ca²⁺]_SR > 130 µM_Cyt, which we consider to be unphysiologically overloaded.

Completing an exhaustive search of the parameter space becomes increasingly computationally intensive as the number of parameters grows. An exhaustive search of an M dimensional space, considering K possible values for each conductance requires K^M evaluations of the cost function. Additionally, as the number of sensors (N) that we use to calculate the cost function increases, the fraction of models tested that are good enough solutions (<I) will decrease. We calculated E(p) for 10⁷ random parameter sets, p, and found only 11 which were GESs (E(p) < 0.05). It is therefore apparent that an exhaustive search is not efficient for finding GES in high dimensional parameter space and for this reason we use a minimization scheme to find GESs. We start by randomly assigning values to each parameter such that 0 < p_i < 3p_i,ref for each i in p, and then minimize them with respect to the cost function, E(p), using the Nelder-Mead simplex algorithm Nelder and Mead (1964) (also known as the Amoeba algorithm) until E(p) < ϵ. Running this procudure 10,000 times yeilds 7263 GESs, with the remaining trials being rejected either because the minimization algorithm does not converge, the SR load constraint is not satis1ed, or the system is found to not be in steady-state (determined by comparing Δ[Ca]_i of the 50th and 51st beat).

The Nelder-Mead Simplex Algorithm

The Nelder-Mead algorithm (Nelder and Mead, 1964) maintains a non-degenerate simplex at each iteration, a geometric 1gure in n dimensions of nonzero volume that is the convex hull of n + 1 vertices, , and their respective function values. Suppose we start from the vector , the simplex can be initialized as , where is a unit vector, and where δ is our guess of the problem’s characteristic length scale. In each iteration, new points are computed, along with their function values, to form a new simplex. The algorithm terminates when the function values at the vertices of the simplex satisfy a predetermined condition. One iteration of the Amoeba algorithm consists of the following steps (the standard values for the coefficients are: α = 1, β = 2, γ = 0.5, σ = 0.5):

1. Order: order and re-label the n + 1 vertices as , such that .Since we want to minimize , we refer to as the best point, to as the worst point, and to as the next worst point. Let refer to the centroid of the n points in the vertex..

2. Reflect: compute the reflected point,. Evaluate if , then obtain a new simplex by replacing the worst point with the reflected point . and go to step 1.

3. Expand: if , compute the expanded point, If , then obtain a new simplex by replacing the worst point with the expanded point and go to step 1; otherwise then obtain a new simplex by replacing the worst point with the reflected point and go to step 1.

4. Contract: At this step, where it is certain that , compute the contracted point .If , obtain a new simplex by replacing the worst point with the expansion point and go to step 1.

5. Shrink: replace all vertices except the best with and go to step 1.

Two sensor search only constraining the Ca2+ transient

The results of the GES search described in the main text were reproduced using only 2 sensors, constraining ⟨[Ca]_i⟩ and Δ[Ca]_i but not constraining [Na]. This results in a broader histogram for the I_NaCa conductance (Figure 3–Figure Supplement 1 compared to Fig. 3).

Acknowledgments

This research was supported by NIH/NHLBI grant 5R01HL114437 and by the Laubisch and Kawata Endowments. The authors thank Yibin Wang for valuable input into this study. A.K. acknowledges stimulating discussions with Eve Marder in the early stage of this work.

Footnotes

• ¹ https://systems.genetics.ucla.edu/about/hmdp2