An ionic model for rhythmic activity in small clusters of embryonic chick ventricular cells

Fast creation of data-driven low-order predictive cardiac tissue excitation models from recorded activation patterns

Excitable systems give rise to important phenomena such as heat waves, epidemics and cardiac arrhythmias. Understanding, forecasting and controlling such systems requires reliable mathematical representations. For cardiac tissue, computational models are commonly generated in a reaction-diffusion framework based on detailed measurements of ionic currents in dedicated single-cell experiments. Here, we show that recorded movies at the tissue-level of stochastic pacing in a single variable are sufficient to generate a mathematical model. Via exponentially weighed moving averages, we create additional state variables, and use simple polynomial regression in the augmented state space to quantify excitation wave dynamics. A spatial gradient-sensing term replaces the classical diffusion as it is more robust to noise. Our pipeline for model creation is demonstrated for an in-silico model and optical voltage mapping recordings of cultured human atrial myocytes and only takes a few minutes. Our findings have the potential for widespread generation, use and on-the-fly refinement of personalised computer models for non-linear phenomena in biology and medicine, such as predictive cardiac digital twins.

Leak current, even with gigaohm seals, can cause misinterpretation of stem cell-derived cardiomyocyte action potential recordings

AIMS

Human-induced pluripotent stem cell-derived cardiomyocytes (iPSC-CMs) have become an essential tool to study arrhythmia mechanisms. Much of the foundational work on these cells, as well as the computational models built from the resultant data, has overlooked the contribution of seal-leak current on the immature and heterogeneous phenotype that has come to define these cells. The aim of this study is to understand the effect of seal-leak current on recordings of action potential (AP) morphology.

METHODS AND RESULTS

Action potentials were recorded in human iPSC-CMs using patch clamp and simulated using previously published mathematical models. Our in silico and in vitro studies demonstrate how seal-leak current depolarizes APs, substantially affecting their morphology, even with seal resistances (Rseal) above 1 GΩ. We show that compensation of this leak current is difficult due to challenges with obtaining accurate measures of Rseal during an experiment. Using simulation, we show that Rseal measures (i) change during an experiment, invalidating the use of pre-rupture values, and (ii) are polluted by the presence of transmembrane currents at every voltage. Finally, we posit that the background sodium current in baseline iPSC-CM models imitates the effects of seal-leak current and is increased to a level that masks the effects of seal-leak current on iPSC-CMs.

CONCLUSION

Based on these findings, we make recommendations to improve iPSC-CM AP data acquisition, interpretation, and model-building. Taking these recommendations into account will improve our understanding of iPSC-CM physiology and the descriptive ability of models built from such data.

The Linkage Phase of the Polymorphism KCNH2-K897T Influences the Electrophysiological Phenotype in hiPSC Models of LQT2

While rare mutations in ion channel genes are primarily responsible for inherited cardiac arrhythmias, common genetic variants are also an important contributor to the clinical heterogeneity observed among mutation carriers. The common single nucleotide polymorphism (SNP) KCNH2-K897T is associated with QT interval duration, but its influence on the disease phenotype in patients with long QT syndrome type 2 (LQT2) remains unclear. Human induced pluripotent stem cells (hiPSCs), coupled with advances in gene editing technologies, are proving an invaluable tool for modeling cardiac genetic diseases and identifying variants responsible for variability in disease expressivity. In this study, we have used isogenic hiPSC-derived cardiomyocytes (hiPSC-CMs) to establish the functional consequences of having the KCNH2-K897T SNP in cis- or trans-orientation with LQT2-causing missense variants either within the pore-loop domain (KCNH2^A561T/WT) or tail region (KCNH2^N996I/WT) of the potassium ion channel, human ether-a-go-go-related gene (hERG). When KCNH2-K897T was on the same allele (cis) as the primary mutation, the hERG channel in hiPSC-CMs exhibited faster activation and deactivation kinetics compared to their trans-oriented counterparts. Consistent with this, hiPSC-CMs with KCNH2-K897T in cis orientation had longer action and field potential durations. Furthermore, there was an increased occurrence of arrhythmic events upon pharmacological blocking of hERG. Collectively, these results indicate that the common polymorphism KCNH2-K897T differs in its influence on LQT2-causing KCNH2 mutations depending on whether it is present in cis or trans. This study corroborates hiPSC-CMs as a powerful platform to investigate the modifying effects of common genetic variants on inherited cardiac arrhythmias and aids in unraveling their contribution to the variable expressivity of these diseases.

Dynamic Clamp in Electrophysiological Studies on Stem Cell-Derived Cardiomyocytes-Why and How?

ABSTRACT

Human pluripotent stem cell-derived cardiomyocytes (hPSC-CMs) are supposed to be a good human-based model, with virtually unlimited cell source, for studies on mechanisms underlying cardiac development and cardiac diseases, and for identification of drug targets. However, a major drawback of hPSC-CMs as a model system, especially for electrophysiological studies, is their depolarized state and associated spontaneous electrical activity. Various approaches are used to overcome this drawback, including the injection of "synthetic" inward rectifier potassium current (IK1), which is computed in real time, based on the recorded membrane potential ("dynamic clamp"). Such injection of an IK1-like current results in quiescent hPSC-CMs with a nondepolarized resting potential that show "adult-like" action potentials on stimulation, with functional availability of the most important ion channels involved in cardiac electrophysiology. These days, dynamic clamp has become a widely appreciated electrophysiological tool. However, setting up a dynamic clamp system can still be laborious and difficult, both because of the required hardware and the implementation of the dedicated software. In the present review, we first summarize the potential mechanisms underlying the depolarized state of hPSC-CMs and the functional consequences of this depolarized state. Next, we explain how an existing manual patch clamp setup can be extended with dynamic clamp. Finally, we shortly validate the extended setup with atrial-like and ventricular-like hPSC-CMs. We feel that dynamic clamp is a highly valuable tool in the field of cellular electrophysiological studies on hPSC-CMs and hope that our directions for setting up such dynamic clamp system may prove helpful.

Multiple Dynamical Mechanisms of Phase-2 Early Afterdepolarizations in a Human Ventricular Myocyte Model: Involvement of Spontaneous SR Ca 2+ Release

Early afterdepolarization (EAD) is known to cause lethal ventricular arrhythmias in long QT syndrome (LQTS). In this study, dynamical mechanisms of EAD formation in human ventricular myocytes (HVMs) were investigated using the mathematical model developed by ten Tusscher and Panfilov (Am J Physiol Heart Circ Physiol 291, 2006). We explored how the rapid (IKr) and slow (IKs) components of delayed-rectifier K+ channel currents, L-type Ca2+ channel current (ICa L), Na+/Ca2+ exchanger current (INCX), and intracellular Ca2+ handling via the sarcoplasmic reticulum (SR) contribute to initiation, termination and modulation of phase-2 EADs during pacing in relation to bifurcation phenomena in non-paced model cells. Parameter-dependent dynamical behaviors of the non-paced model cell were determined by calculating stabilities of equilibrium points (EPs) and limit cycles, and bifurcation points to construct bifurcation diagrams. Action potentials (APs) and EADs during pacing were reproduced by numerical simulations for constructing phase diagrams of the paced model cell dynamics. Results are summarized as follows: (1) A modified version of the ten Tusscher-Panfilov model with accelerated ICaL inactivation could reproduce bradycardia-related EADs in LQTS type 2 and β-adrenergic stimulation-induced EADs in LQTS type 1. (2) Two types of EADs with different initiation mechanisms, ICaL reactivation-dependent and spontaneous SR Ca2+ release-mediated EADs, were detected. (3) Termination of EADs (AP repolarization) during pacing depended on the slow activation of IKs. (4) Spontaneous SR Ca2+ releases occurred at higher Ca2+ uptake rates, attributable to the instability of steady-state intracellular Ca2+ concentrations. Dynamical mechanisms of EAD formation and termination in the paced model cell are closely related to stability changes (bifurcations) in dynamical behaviors of the non-paced model cell, but they are model-dependent. Nevertheless, the modified ten Tusscher-Panfilov model would be useful for systematically investigating possible dynamical mechanisms of EAD-related arrhythmias in LQTS.

Cell Culture Platforms with Controllable Stiffness for Chick Embryonic Cardiomyocytes

For several years, cell culture techniques have been physiologically relevant to understand living organisms both structurally and functionally, aiming at preserving as carefully as possible the in vivo integrity and function of the cells. However, when studying cardiac cells, glass or plastic Petri dishes and culture-coated plates lack important cues that do not allow to maintain the desired phenotype, especially for primary cell culture. In this work, we show that microscaffolds made with polydimethylsiloxane (PDMS) enable modulating the stiffness of the surface of the culture substrate and this originates different patterns of adhesion, self-organization, and synchronized or propagated activity in the culture of chick embryonic cardiomyocytes. Thanks to the calcium imaging technique, we found that the substrate stiffness affected cardiomyocyte adhesion, as well as the calcium signal propagation in the formed tissue. The patterns of activity shown by the calcium fluorescence variations are reliable clues of the functional organization achieved by the cell layers. We found that PDMS substrates with a stiffness of 25 kPa did not allow the formation of cell layers and therefore the optimal propagation of the intracellular calcium signals, while softer PDMS substrates with Young’s modulus within the physiological in vivo reported range did permit synchronized and coordinated contractility and intracellular calcium activity. This type of methodology allows us to study phenomena such as arrhythmias. For example, the occurrence of synchronized activity or rotors that can initiate or maintain cardiac arrhythmias can be reproduced on different substrates for study, so that replacement tissues or patches can be better designed.

Regularity of beating of small clusters of embryonic chick ventricular heart-cells: experiment vs. stochastic single-channel population model

The transmembrane potential is recorded from small isopotential clusters of 2-4 embryonic chick ventricular cells spontaneously generating action potentials. We analyze the cycle-to-cycle fluctuations in the time between successive action potentials (the interbeat interval or IBI). We also convert an existing model of electrical activity in the cluster, which is formulated as a Hodgkin-Huxley-like deterministic system of nonlinear ordinary differential equations describing five individual ionic currents, into a stochastic model consisting of a population of ∼20 000 independently and randomly gating ionic channels, with the randomness being set by a real physical stochastic process (radio static). This stochastic model, implemented using the Clay-DeFelice algorithm, reproduces the fluctuations seen experimentally: e.g., the coefficient of variation (standard deviation/mean) of IBI is 4.3% in the model vs. the 3.9% average value of the 17 clusters studied. The model also replicates all but one of several other quantitative measures of the experimental results, including the power spectrum and correlation integral of the voltage, as well as the histogram, Poincaré plot, serial correlation coefficients, power spectrum, detrended fluctuation analysis, approximate entropy, and sample entropy of IBI. The channel noise from one particular ionic current (I_Ks), which has channel kinetics that are relatively slow compared to that of the other currents, makes the major contribution to the fluctuations in IBI. Reproduction of the experimental coefficient of variation of IBI by adding a Gaussian white noise-current into the deterministic model necessitates using an unrealistically high noise-current amplitude. Indeed, a major implication of the modelling results is that, given the wide range of time-scales over which the various species of channels open and close, only a cell-specific stochastic model that is formulated taking into consideration the widely different ranges in the frequency content of the channel-noise produced by the opening and closing of several different types of channels will be able to reproduce precisely the various effects due to membrane noise seen in a particular electrophysiological preparation.

Dynamical mechanisms of phase-2 early afterdepolarizations in human ventricular myocytes: insights from bifurcation analyses of two mathematical models

Early afterdepolarization (EAD) is known as a cause of ventricular arrhythmias in long QT syndromes. We theoretically investigated how the rapid (I_Kr) and slow (I_Ks) components of delayed-rectifier K⁺ channel currents, L-type Ca²⁺ channel current (I_CaL), Na⁺/Ca²⁺ exchanger current (I_NCX), Na⁺-K⁺ pump current (I_NaK), intracellular Ca²⁺ (Ca_i) handling via sarcoplasmic reticulum (SR), and intracellular Na⁺ concentration (Na_i) contribute to initiation, termination, and modulation of phase-2 EADs, using two human ventricular myocyte models. Bifurcation structures of dynamical behaviors in model cells were explored by calculating equilibrium points, limit cycles (LCs), and bifurcation points as functions of parameters. EADs were reproduced by numerical simulations. The results are summarized as follows: 1) decreasing I_Ks and/or I_Kr or increasing I_CaL led to EAD generation, to which mid-myocardial cell models were especially susceptible; the parameter regions of EADs overlapped the regions of stable LCs. 2) Two types of EADs (termination mechanisms), I_Ks activation-dependent and I_CaL inactivation-dependent EADs, were detected; I_Ks was not necessarily required for EAD formation. 3) Inhibiting I_NCX suppressed EADs via facilitating Ca²⁺-dependent I_CaL inactivation. 4) Ca_i dynamics (SR Ca²⁺ handling) and Na_i strongly affected bifurcations and EAD generation in model cells via modulating I_CaL, I_NCX, and I_NaK Parameter regions of EADs, often overlapping those of stable LCs, shifted depending on Ca_i and Na_i in stationary and dynamic states. 5) Bradycardia-related induction of EADs was mainly due to decreases in Na_i at lower pacing rates. This study demonstrates that bifurcation analysis allows us to understand the dynamical mechanisms of EAD formation more profoundly.

NEW & NOTEWORTHY

We investigated mechanisms of phase-2 early afterdepolarization (EAD) by bifurcation analyses of human ventricular myocyte (HVM) models. EAD formation in paced HVMs basically depended on bifurcation phenomena in non-paced HVMs, but was strongly affected by intracellular ion concentrations in stationary and dynamic states. EAD generation did not necessarily require I_Ks.

Cell-specific cardiac electrophysiology models

The traditional cardiac model-building paradigm involves constructing a composite model using data collected from many cells. Equations are derived for each relevant cellular component (e.g., ion channel, exchanger) independently. After the equations for all components are combined to form the composite model, a subset of parameters is tuned, often arbitrarily and by hand, until the model output matches a target objective, such as an action potential. Unfortunately, such models often fail to accurately simulate behavior that is dynamically dissimilar (e.g., arrhythmia) to the simple target objective to which the model was fit. In this study, we develop a new approach in which data are collected via a series of complex electrophysiology protocols from single cardiac myocytes and then used to tune model parameters via a parallel fitting method known as a genetic algorithm (GA). The dynamical complexity of the electrophysiological data, which can only be fit by an automated method such as a GA, leads to more accurately parameterized models that can simulate rich cardiac dynamics. The feasibility of the method is first validated computationally, after which it is used to develop models of isolated guinea pig ventricular myocytes that simulate the electrophysiological dynamics significantly better than does a standard guinea pig model. In addition to improving model fidelity generally, this approach can be used to generate a cell-specific model. By so doing, the approach may be useful in applications ranging from studying the implications of cell-to-cell variability to the prediction of intersubject differences in response to pharmacological treatment.

Mathematical models of cardiac cell electrophysiology are widely used as predictive and illuminatory tools, but have been developed for decades using a suboptimal process. The models are typically constructed by manual adjustment of parameters to fit simple data and therefore often underperform when used to predict complex behavior such as arrhythmias. We present a novel method of model parameterization using automated optimization and dynamically rich fitting data and then demonstrate that this approach is better at finding the “real” model of a cell. Application of the method to cardiac myocytes leads to cell-specific models, which may enable well-controlled studies of both cellular- and subject-level population heterogeneity in disease propensity and response to therapies.

Alterations of field potentials in isotropic cardiomyocyte cell layers induced by multiple endogenous pacemakers under normal and hypothermal conditions

The use of autonomous contracting randomly grown cardiomyocyte monolayers cultivated on microelectrode arrays (MEAs) represents an accepted experimental setting for preclinical experimental research in the field of cardiac electrophysiology. A dominant pacemaker forces a monolayer to adhere to a regular and synchronized contraction. Randomly distributed multiple pacemakers interfere with this dominant center, resulting in more or less frequent changes of propagation direction. This study aims to characterize the impact of changing propagation directions at single electrodes of the MEA on the four intrinsic parameters of registered field potentials (FPs) FPrise, FPMIN, FPpre, and FPdur and conduction velocity (CV) under normal and hypothermal conditions. Primary cultures of chicken cardiomyocytes ( n = 18) were plated directly onto MEAs and FPs were recorded in a temperature range between 37 and 29°C. The number and spatiotemporal distribution of biological and artificial pacemakers of each cell layer inside and outside of the MEA registration area were evaluated using an algorithm developed in-house. In almost every second myocardial cell layer, interfering autonomous pacemakers were detected at stable temperatures, showing random spatial distributions with similar beating rates. Additionally, a temperature-dependent change of the dominant pacemaker center was observed in n = 16 experiments. A significant spread-direction-dependent variation of CV, FPrise, FPMIN, and FPpre up to 14% could be measured between different endogenous pacemakers. In conclusion, based on our results, disregarding the spatial origin of excitation may lead to misinterpretations and erroneous conclusions of FP parameters in the verification of research hypotheses in cellular electrocardiology.

Modern perspectives on numerical modeling of cardiac pacemaker cell

Cardiac pacemaking is a complex phenomenon that is still not completely understood. Together with experimental studies, numerical modeling has been traditionally used to acquire mechanistic insights in this research area. This review summarizes the present state of numerical modeling of the cardiac pacemaker, including approaches to resolve present paradoxes and controversies. Specifically we discuss the requirement for realistic modeling to consider symmetrical importance of both intracellular and cell membrane processes (within a recent "coupled-clock" theory). Promising future developments of the complex pacemaker system models include the introduction of local calcium control, mitochondria function, and biochemical regulation of protein phosphorylation and cAMP production. Modern numerical and theoretical methods such as multi-parameter sensitivity analyses within extended populations of models and bifurcation analyses are also important for the definition of the most realistic parameters that describe a robust, yet simultaneously flexible operation of the coupled-clock pacemaker cell system. The systems approach to exploring cardiac pacemaker function will guide development of new therapies such as biological pacemakers for treating insufficient cardiac pacemaker function that becomes especially prevalent with advancing age.

Beat to beat variability of embryonic chick heart cells under septic conditions: application and evaluation of entropy as well as fractal measures

Extracardiac factors of heart rate variability have commonly been investigated using linear and nonlinear methods for a long time. Recently, intracardiac mechanisms on an electrophysiological basis have been found to be also important. This work is focused on the evaluation of complex measures of temporal signals gained with microelectrode measurements of embryonic chick heart aggregates. Septic conditions were mimicked in vitro by lipopolysaccharide (LPS) administration in order to investigate the influence on beat to beat variability. Surrogate data analysis revealed high statistical significances for normalized complexity measures.

Modeling gender effects on electrical activity of single ventricular myocytes

In this study we investigate the mechanisms underlying gender differences in the generation of arrhythmias in the long QT and Brugada syndromes. Simulations were conducted at the single myocyte level using a detailed mathematical model of human ventricular myocytes. Given the scarce human data on the gender-related differences in single cardiac cells, we assumed gender-related differences in five ionic-current systems: fast sodium current (INa), slowly inactivating late sodium current (INal), transient outward potassium current (Ito), slow delayed rectifier potassium current (IKs), and calcium current through the L-type channel (ICa(L)), based on experimental results obtained in canine myocytes. Our modeling results suggest that in left ventricular myocytes, enhanced INal under conditions of reduced repolarization reserve results in sex-dependent development of early afterdepolarizations (EADs) in the post-pause action potentials (APs). Moreover, this modeling study demonstrates increased propensity for the development of the loss of the AP dome in male epicardial myocytes of the right ventricle compared with other types of myocytes from the left and right ventricles. Finally, we also found a slight effect of INal on gender-dependent loss of AP dome in epicardial right ventricular myocytes. In conclusion, at the cellular level, gender differences in the development of EADs and the propensity to develop the loss of the AP dome can be attributed to male/female related differences in INa, INal, Ito, IKs, and ICa(L).

Effect of hyperpolarization-activated current I(f) on robustness of sinoatrial node pacemaking: theoretical study on influence of intracellular Na(+) concentration

To elucidate the effects of hyperpolarization-activated current I(f) on robustness of sinoatrial node (SAN) pacemaking in connection with intracellular Na(+) concentration (Na(i)) changes, we theoretically investigated 1) the impacts of I(f) on dynamical properties of SAN model cells during inhibition of L-type Ca(2+) channel currents (I(CaL)) or hyperpolarizing loads and 2) I(f)-dependent changes in Na(i) and their effects on dynamical properties of model cells. Bifurcation analyses were performed for Na(i)-variable and Na(i)-fixed versions of mathematical models for rabbit SAN cells; equilibrium points (EPs), limit cycles (LCs), and their stability were determined as functions of model parameters. Increasing I(f) conductance (g(f)) shrank I(CaL) conductance (g(CaL)) regions of unstable EPs and stable LCs (rhythmic firings) in the Na(i)-variable system but slightly broadened that of rhythmic firings at lower g(f) in the Na(i)-fixed system. In the Na(i)-variable system, increased g(f) yielded elevations in Na(i) at EPs and during spontaneous oscillations, which caused EP stabilization and shrinkage in the parameter regions of unstable EPs and rhythmic firings. As g(f) increased, parameter regions of unstable EPs and stable LCs determined for hyperpolarizing loads shrank in the Na(i)-variable system but were enlarged in the Na(i)-fixed system. These findings suggest that 1) I(f) does not enhance but rather attenuates robustness of rabbit SAN cells via facilitating EP stabilization and LC destabilization even in physiological g(f) ranges; and 2) the enhancing effect of I(f) on robustness of pacemaker activity, which could be observed at lower g(f) when Na(i) was fixed, is actually reversed by I(f)-dependent changes in Na(i).

Mechanism-based facilitated maturation of human pluripotent stem cell-derived cardiomyocytes

BACKGROUND

Human embryonic stem cells (hESCs) can be efficiently and reproducibly directed into cardiomyocytes (CMs) using stage-specific induction protocols. However, their functional properties and suitability for clinical and other applications have not been evaluated.

METHODS AND RESULTS

Here we showed that CMs derived from multiple pluripotent human stem cell lines (hESC: H1, HES2) and types (induced pluripotent stem cell) using different in vitro differentiation protocols (embryoid body formation, endodermal induction, directed differentiation) commonly displayed immature, proarrhythmic action potential properties such as high degree of automaticity, depolarized resting membrane potential, Phase 4- depolarization, and delayed after-depolarization. Among the panoply of sarcolemmal ionic currents investigated (I(Na)(+)/I(CaL)(+)/I(Kr)(+)/I(NCX)(+)/I(f)(+)/I(to)(+)/I(K1)(-)/I(Ks)(-)), we pinpointed the lack of the Kir2.1-encoded inwardly rectifying K(+) current (I(K1)) as the single mechanistic contributor to the observed immature electrophysiological properties in hESC-CMs. Forced expression of Kir2.1 in hESC-CMs led to robust expression of Ba(2+)-sensitive I(K1) and, more importantly, completely ablated all the proarrhythmic action potential traits, rendering the electrophysiological phenotype indistinguishable from the adult counterparts. These results provided the first link of a complex developmentally arrested phenotype to a major effector gene, and importantly, further led us to develop a bio-mimetic culturing strategy for enhancing maturation.

CONCLUSIONS

By providing the environmental cues that are missing in conventional culturing method, this approach did not require any genetic or pharmacological interventions. Our findings can facilitate clinical applications, drug discovery, and cardiotoxicity screening by improving the yield, safety, and efficacy of derived CMs.

Roles of sarcoplasmic reticulum Ca2+ cycling and Na+/Ca2+ exchanger in sinoatrial node pacemaking: Insights from bifurcation analysis of mathematical models

To elucidate the roles of sarcoplasmic reticulum (SR) Ca2+ cycling and Na+/Ca2+ exchanger (NCX) in sinoatrial node (SAN) pacemaking, we have applied stability and bifurcation analyses to a coupled-clock system model developed by Maltsev and Lakatta ( Am J Physiol Heart Circ Physiol 296: H594-H615, 2009). Equilibrium point (EP) at which the system is stationary (i.e., the oscillatory system fails to function), periodic orbit (limit cycle), and their stability were determined as functions of model parameters. The stability analysis to detect bifurcation points confirmed crucial importance of SR Ca2+ pumping rate constant ( Pup), NCX density ( kNCX), and L-type Ca2+ channel conductance for the system function reported in previous parameter-dependent numerical simulations. We showed, however, that the model cell does not exhibit self-sustained automaticity of SR Ca2+ release at any clamped voltage and therefore needs further tuning to reproduce oscillatory local Ca2+ release and net membrane current reported experimentally at −10 mV. Our further extended bifurcation analyses revealed important novel features of the pacemaker system that go beyond prior numerical simulations in relation to the roles of SR Ca2+ cycling and NCX in SAN pacemaking. Specifically, we found that 1) NCX contributes to EP instability and enhancement of robustness in the full system during normal spontaneous action potential firings, while stabilizing EPs to prevent sustained Ca2+ oscillations under voltage clamping; 2) SR requires relatively large kNCX and subsarcolemmal Ca2+ diffusion barrier (i.e., subspace) to contribute to EP destabilization and enhancement of robustness; and 3) decrementing Pup or kNCX decreased the full system robustness against hyperpolarizing loads because EP stabilization and cessation of pacemaking were observed at the lower critical amplitude of hyperpolarizing bias currents, suggesting that SR Ca2+ cycling contributes to enhancement of the full system robustness by modulating NCX currents and promoting EP destabilization.

A mathematical model of action potentials of mouse sinoatrial node cells with molecular bases

Genetically modified mice are popular experimental models for studying the molecular bases and mechanisms of cardiac arrhythmia. A postgenome challenge is to classify the functional roles of genes in cardiac function. To unveil the functional role of various genetic isoforms of ion channels in generating cardiac pacemaking action potentials (APs), a mathematical model for spontaneous APs of mouse sinoatrial node (SAN) cells was developed. The model takes into account the biophysical properties of membrane ionic currents and intracellular mechanisms contributing to spontaneous mouse SAN APs. The model was validated by its ability to reproduce the physiological exceptionally short APs and high pacing rates of mouse SAN cells. The functional roles of individual membrane currents were evaluated by blocking their coding channels. The roles of intracellular Ca²⁺-handling mechanisms on cardiac pacemaking were also investigated in the model. The robustness of model pacemaking behavior was evaluated by means of one- and two-parameter analyses in wide parameter value ranges. This model provides a predictive tool for cellular level outcomes of electrophysiological experiments. It forms the basis for future model development and further studies into complex pacemaking mechanisms as more quantitative experimental data become available.

Unstable eigenmodes are possible drivers for cardiac arrhythmias

The well-organized contraction of each heartbeat is enabled by an electrical wave traversing and exciting the myocardium in a regular manner. Perturbations to this wave, referred to as arrhythmias, can lead to lethal fibrillation if not treated within minutes. One manner in which arrhythmias originate is an ill-fated interaction of the regular electrical signal controlling the heartbeat, the sinus wave, with an ectopic stimulus. It is not fully understood how and when ectopic waves are generated. Based on mathematical models, we show that ectopic beats can be characterized in terms of unstable eigenmodes of the resting state.

Regression analysis for constraining free parameters in electrophysiological models of cardiac cells

A major challenge in computational biology is constraining free parameters in mathematical models. Adjusting a parameter to make a given model output more realistic sometimes has unexpected and undesirable effects on other model behaviors. Here, we extend a regression-based method for parameter sensitivity analysis and show that a straightforward procedure can uniquely define most ionic conductances in a well-known model of the human ventricular myocyte. The model's parameter sensitivity was analyzed by randomizing ionic conductances, running repeated simulations to measure physiological outputs, then collecting the randomized parameters and simulation results as "input" and "output" matrices, respectively. Multivariable regression derived a matrix whose elements indicate how changes in conductances influence model outputs. We show here that if the number of linearly-independent outputs equals the number of inputs, the regression matrix can be inverted. This is significant, because it implies that the inverted matrix can specify the ionic conductances that are required to generate a particular combination of model outputs. Applying this idea to the myocyte model tested, we found that most ionic conductances could be specified with precision (R(2) > 0.77 for 12 out of 16 parameters). We also applied this method to a test case of changes in electrophysiology caused by heart failure and found that changes in most parameters could be well predicted. We complemented our findings using a Bayesian approach to demonstrate that model parameters cannot be specified using limited outputs, but they can be successfully constrained if multiple outputs are considered. Our results place on a solid mathematical footing the intuition-based procedure simultaneously matching a model's output to several data sets. More generally, this method shows promise as a tool to define model parameters, in electrophysiology and in other biological fields.

Roles of hyperpolarization-activated current If in sinoatrial node pacemaking: insights from bifurcation analysis of mathematical models

To elucidate the roles of hyperpolarization-activated current (I(f)) in sinoatrial node (SAN) pacemaking, we theoretically investigated 1) the effects of I(f) on stability and bifurcation during hyperpolarization of SAN cells; 2) combined effects of I(f) and the sustained inward current (I(st)) or Na(+) channel current (I(Na)) on robustness of pacemaking against hyperpolarization; and 3) whether blocking I(f) abolishes pacemaker activity under certain conditions. Bifurcation analyses were performed for mathematical models of rabbit SAN cells; equilibrium points (EPs), periodic orbits, and their stability were determined as functions of parameters. Unstable steady-state potential region determined with applications of constant bias currents shrunk as I(f) density increased. In the central SAN cell, the critical acetylcholine concentration at which bifurcations, to yield a stable EP and quiescence, occur was increased by smaller I(f), but decreased by larger I(f). In contrast, the critical acetylcholine concentration and conductance of gap junctions between SAN and atrial cells at bifurcations progressively increased with enhancing I(f) in the peripheral SAN cell. These effects of I(f) were significantly attenuated by eliminating I(st) or I(Na), or by accelerating their inactivation. Under hyperpolarized conditions, blocking I(f) abolished SAN pacemaking via bifurcations. These results suggest that 1) I(f) itself cannot destabilize EPs; 2) I(f) improves SAN cell robustness against parasympathetic stimulation via preventing bifurcations in the presence of I(st) or I(Na); 3) I(f) dramatically enhances peripheral cell robustness against electrotonic loads of the atrium in combination with I(Na); and 4) pacemaker activity of hyperpolarized SAN cells could be abolished by blocking I(f).

Uniqueness and stability of action potential models during rest, pacing, and conduction using problem-solving environment

Development and application of physiologically detailed dynamic models of the action potential (AP) and Ca2+ cycling in cardiac cells is a rapidly growing aspect of computational cardiac electrophysiology. Given the large scale of the nonlinear system involved, questions were recently raised regarding reproducibility, numerical stability, and uniqueness of model solutions, as well as ability of the model to simulate AP propagation in multicellular configurations. To address these issues, we reexamined ventricular models of myocyte AP developed in our laboratory with the following results. 1), Recognizing that the model involves a system of differential-algebraic equations, a procedure is developed for estimating consistent initial conditions that insure uniqueness and stability of the solution. 2), Model parameters that can be used to modify these initial conditions according to experimental values are identified. 3), A convergence criterion for steady-state solution is defined based on tracking the incremental contribution of each ion species to the membrane voltage. 4), Singularities in state variable formulations are removed analytically. 5), A biphasic current stimulus is implemented to completely eliminate stimulus artifact during long-term pacing over a broad range of frequencies. 6), Using the AP computed based on 1-5 above, an efficient scheme is developed for computing propagation in multicellular models.

Stochastic and spatial influences on drug-induced bifurcations in cardiac tissue culture

The addition of a drug that specifically blocks a potassium channel in spontaneously beating aggregates of chick heart cells leads to complex bifurcations over time. A stochastic partial differential equation model based on discrete ionic currents recorded in these cells demonstrates that drug diffusion and noise can induce the coupled beats and bursting rhythms observed. These results provide further evidence that stochastic events at a subcellular level are needed to understand complex cardiac arrhythmias and play an important role in the onset of these arrhythmias.

Synergism of coupled subsarcolemmal Ca2+ clocks and sarcolemmal voltage clocks confers robust and flexible pacemaker function in a novel pacemaker cell model

Recent experimental studies have demonstrated that sinoatrial node cells (SANC) generate spontaneous, rhythmic, local subsarcolemmal Ca(2+) releases (Ca(2+) clock), which occur during late diastolic depolarization (DD) and interact with the classic sarcolemmal voltage oscillator (membrane clock) by activating Na(+)-Ca(2+) exchanger current (I(NCX)). This and other interactions between clocks, however, are not captured by existing essentially membrane-delimited cardiac pacemaker cell numerical models. Using wide-scale parametric analysis of classic formulations of membrane clock and Ca(2+) cycling, we have constructed and initially explored a prototype rabbit SANC model featuring both clocks. Our coupled oscillator system exhibits greater robustness and flexibility than membrane clock operating alone. Rhythmic spontaneous Ca(2+) releases of sarcoplasmic reticulum (SR)-based Ca(2+) clock ignite rhythmic action potentials via late DD I(NCX) over much broader ranges of membrane clock parameters [e.g., L-type Ca(2+) current (I(CaL)) and/or hyperpolarization-activated ("funny") current (I(f)) conductances]. The system Ca(2+) clock includes SR and sarcolemmal Ca(2+) fluxes, which optimize cell Ca(2+) balance to increase amplitudes of both SR Ca(2+) release and late DD I(NCX) as SR Ca(2+) pumping rate increases, resulting in a broad pacemaker rate modulation (1.8-4.6 Hz). In contrast, the rate modulation range via membrane clock parameters is substantially smaller when Ca(2+) clock is unchanged or lacking. When Ca(2+) clock is disabled, the system parametric space for fail-safe SANC operation considerably shrinks: without rhythmic late DD I(NCX) ignition signals membrane clock substantially slows, becomes dysrhythmic, or halts. In conclusion, the Ca(2+) clock is a new critical dimension in SANC function. A synergism of the coupled function of Ca(2+) and membrane clocks confers fail-safe SANC operation at greatly varying rates.

Rhythms of high-grade block in an ionic model of a strand of regionally ischemic ventricular muscle

Electrical alternans, a beat-to-beat alternation in the electrocardiogram or electrogram, is frequently seen during the first few minutes of acute myocardial ischemia, and is often immediately followed by malignant cardiac arrhythmias such as ventricular tachycardia and ventricular fibrillation. As ischemia progresses, higher-order periodic rhythms (e.g., period-4) can replace the period-2 alternans rhythm. This is also seen in modelling work on a two-dimensional (2-D) sheet of regionally ischemic ventricular muscle. In addition, in the experimental work, ventricular arrhythmias are overwhelmingly seen only after the higher-order rhythms arise. We investigate an ionic model of a strand of ischemic ventricular muscle, constructed as a 3-cm-long 1-D cable with a centrally located 1-cm-long segment exposed to an elevated extracellular potassium concentration ([K(+)](o)). As [K(+)](o) is raised in this "ischemic segment" to represent one major effect of ongoing ischemia, the sequence of rhythms {1:1-->2:2 (alternans)-->2:1} is seen. With further increase in [K(+)](o), one sees higher-order periodic 2N:M rhythms {2:1-->4:2-->4:1-->6:2-->6:1-->8:2-->8:1}. In a 2N:M cycle, only M of the 2N action potentials generated at the proximal end of the cable successfully traverse the ischemic segment, with the remaining ones being blocked within the ischemic segment. Finally, there is a transition to complete block {8:1-->2:0-->1:0} (in an n:0 rhythm, all action potentials die out within the ischemic segment). Changing the length of the ischemic segment results in different rhythms and transitions being seen: e.g., when the ischemic segment is 2 cm long, the period-6 rhythms are not seen; when it is 0.5 cm long, there is a 3:1 rhythm interposed between the 2:1 and 1:0 rhythms. We discuss the relevance of our results to the experimental observations on the higher-order rhythms that presage reentrant ischemic ventricular arrhythmias.

Computer modelling of the sinoatrial node

Over the past decades patch-clamp experiments have provided us with detailed information on the different types of ion channels that are present in the cardiac cell membrane. Sophisticated cardiac cell models based on these data can help us understand how the different types of ion channels act together to produce the cardiac action potential. In the field of biological pacemaker engineering, such models provide important instruments for the assessment of the functional implications of changes in density of specific ion channels aimed at producing stable pacemaker activity. In this review, an overview is given of the progress made in cardiac cell modelling, with particular emphasis on the development of sinoatrial (SA) nodal cell models. Also, attention is given to the increasing number of publicly available tools for non-experts in computer modelling to run cardiac cell models.

Embryonic cardiac arrhythmia and generation of reactive oxygen species: common teratogenic mechanism for IKr blocking drugs

In the adult organism, it is well established that hypoxia followed by reperfusion may be fatal and result in generation of reactive oxygen species (ROS) and subsequent tissue damage. There is also considerable evidence that temporary decrease or interruption in oxygen supply to the embryo and ROS generation during reperfusion result in tissue damage in embryonic tissues. A wide spectrum of different malformations by transient embryonic hypoxia could be produced, depending on the duration, extent, and timing of the hypoxic event. It is the contention of this paper that drugs that block the potassium channel IKr, either as an intended pharmacologic effect or as an unwanted side-effect, are potentially teratogenic by a common ROS related mechanism. Drugs blocking the IKr channel, such as almokalant, dofetilide, phenytoin, cisapride and astemizole, do all produce a similar pattern of hypoxia-related malformations. Mechanistic studies show that the malformations are preceded by embryonic cardiac arrhythmia and periods of hypoxia/reoxygenation in embryonic tissues. Pretreatment or simultaneous treatment with radical scavengers with capacity to capture ROS, markedly decrease the teratogenicity of different IKr blocking drugs. A second aim of this review is to demonstrate that the conventional design of teratology studies is not optimal to detect malformations caused by IKr blocking drugs. Repeated high doses result in high incidences of embryonic death due embryonic cardiac arrhythmia, thus masking their teratogenic potential. Instead, single dosing on specific days is proposed to be a better way to characterize the teratogenic potential of Ikr blocking drugs.

Effects of pacemaker currents on creation and modulation of human ventricular pacemaker: theoretical study with application to biological pacemaker engineering

A cardiac biological pacemaker (BP) has been created by suppression of the inward rectifier K+ current ( IK1) or overexpression of the hyperpolarization-activated current ( Ih). We theoretically investigated the effects of incorporating Ih, T-type Ca2+ current ( ICa,T), sustained inward current ( Ist), and/or low-voltage-activated L-type Ca2+ channel current ( ICa,LD) on 1) creation of BP cells, 2) robustness of BP activity to electrotonic loads of nonpacemaking (NP) cells, and 3) BP cell ability to drive NP cells. We used a single-cell model for human ventricular myocytes (HVMs) and also coupled-cell models composed of BP and NP cells. Bifurcation structures of the model cells were explored during changes in conductance of the currents and gap junction. Incorporating the pacemaker currents did not yield BP activity in HVM with normal IK1 but increased the critical IK1 conductance for BP activity to emerge. Expressing Ih appeared to be most helpful in facilitating creation of BP cells via IK1 suppression. In the coupled-cell model, Ist significantly enlarged the gap conductance ( GC) region where stable BP cell pacemaking and NP cell driving occur, reducing the number of BP cells required for robust pacemaking and driving. In contrast, Ih enlarged the GC region of pacemaking and driving only when IK1 of the NP cell was relatively low. ICa,T or ICa,LD exerted effects similar to those of Ist but caused shrinkage or irregularity of BP oscillations. These findings suggest that expressing Ist most effectively improves the structural stability of BPs to electrotonic loads and the BP ability to drive the ventricle.

Computational biology in the study of cardiac ion channels and cell electrophysiology

The cardiac cell is a complex biological system where various processes interact to generate electrical excitation (the action potential, AP) and contraction. During AP generation, membrane ion channels interact nonlinearly with dynamically changing ionic concentrations and varying transmembrane voltage, and are subject to regulatory processes. In recent years, a large body of knowledge has accumulated on the molecular structure of cardiac ion channels, their function, and their modification by genetic mutations that are associated with cardiac arrhythmias and sudden death. However, ion channels are typically studied in isolation (in expression systems or isolated membrane patches), away from the physiological environment of the cell where they interact to generate the AP. A major challenge remains the integration of ion-channel properties into the functioning, complex and highly interactive cell system, with the objective to relate molecular-level processes and their modification by disease to whole-cell function and clinical phenotype. In this article we describe how computational biology can be used to achieve such integration. We explain how mathematical (Markov) models of ion-channel kinetics are incorporated into integrated models of cardiac cells to compute the AP. We provide examples of mathematical (computer) simulations of physiological and pathological phenomena, including AP adaptation to changes in heart rate, genetic mutations in SCN5A and HERG genes that are associated with fatal cardiac arrhythmias, and effects of the CaMKII regulatory pathway and beta-adrenergic cascade on the cell electrophysiological function.

Dynamical mechanisms of pacemaker generation in IK1-downregulated human ventricular myocytes: insights from bifurcation analyses of a mathematical model

Dynamical mechanisms of the biological pacemaker (BP) generation in human ventricular myocytes were investigated by bifurcation analyses of a mathematical model. Equilibrium points (EPs), periodic orbits, stability of EPs, and bifurcation points were determined as functions of bifurcation parameters, such as the maximum conductance of inward-rectifier K+ current (I(K1)), for constructing bifurcation diagrams. Stable limit cycles (BP activity) abruptly appeared around an unstable EP via a saddle-node bifurcation when I(K1) was suppressed by 84.6%. After the bifurcation at which a stable EP disappears, the I(K1)-reduced system has an unstable EP only, which is essentially important for stable pacemaking. To elucidate how individual sarcolemmal currents contribute to EP instability and BP generation, we further explored the bifurcation structures of the system during changes in L-type Ca2+ channel current (I(Ca,L)), delayed-rectifier K+ currents (I(K)), or Na(+)/Ca2+ exchanger current (I(NaCa)). Our results suggest that 1), I(Ca,L) is, but I(K) or I(NaCa) is not, responsible for EP instability as a requisite to stable BP generation; 2), I(K) is indispensable for robust pacemaking with large amplitude, high upstroke velocity, and stable frequency; and 3), I(NaCa) is the dominant pacemaker current but is not necessarily required for the generation of spontaneous oscillations.