Calcium alternans in a couplon network model of ventricular myocytes: role of sarcoplasmic reticulum load

Diversity of cells and signals in the cardiovascular system

This white paper is the outcome of the seventh UC Davis Cardiovascular Research Symposium on Systems Approach to Understanding Cardiovascular Disease and Arrhythmia. This biannual meeting aims to bring together leading experts in subfields of cardiovascular biomedicine to focus on topics of importance to the field. The theme of the 2022 Symposium was 'Cell Diversity in the Cardiovascular System, cell-autonomous and cell-cell signalling'. Experts in the field contributed their experimental and mathematical modelling perspectives and discussed emerging questions, controversies, and challenges in examining cell and signal diversity, co-ordination and interrelationships involved in cardiovascular function. This paper originates from the topics of formal presentations and informal discussions from the Symposium, which aimed to develop a holistic view of how the multiple cell types in the cardiovascular system integrate to influence cardiovascular function, disease progression and therapeutic strategies. The first section describes the major cell types (e.g. cardiomyocytes, vascular smooth muscle and endothelial cells, fibroblasts, neurons, immune cells, etc.) and the signals involved in cardiovascular function. The second section emphasizes the complexity at the subcellular, cellular and system levels in the context of cardiovascular development, ageing and disease. Finally, the third section surveys the technological innovations that allow the interrogation of this diversity and advancing our understanding of the integrated cardiovascular function and dysfunction.

Calcium- and voltage-driven atrial alternans: Insight from [Ca]i and Vm asynchrony

Cardiac alternans is defined as beat-to-beat alternations in contraction strength, action potential duration (APD), and Ca transient (CaT) amplitude. Cardiac excitation-contraction coupling relies on the activity of two bidirectionally coupled excitable systems, membrane voltage (Vm ) and Ca release. Alternans has been classified as Vm - or Ca-driven, depending whether a disturbance of Vm or [Ca]i regulation drives the alternans. We determined the primary driver of pacing induced alternans in rabbit atrial myocytes, using combined patch clamp and fluorescence [Ca]i and Vm measurements. APD and CaT alternans are typically synchronized; however, uncoupling between APD and CaT regulation can lead to CaT alternans in the absence of APD alternans, and APD alternans can fail to precipitate CaT alternans, suggesting a considerable degree of independence of CaT and APD alternans. Using alternans AP voltage clamp protocols with extra APs showed that most frequently the pre-existing CaT alternans pattern prevailed after the extra-beat, indicating that alternans is Ca-driven. In electrically coupled cell pairs, dyssynchrony of APD and CaT alternans points to autonomous regulation of CaT alternans. Thus, with three novel experimental protocols, we collected evidence for Ca-driven alternans; however, the intimately intertwined regulation of Vm and [Ca]i precludes entirely independent development of CaT and APD alternans.

Cardiac Alternans: From Bedside to Bench and Back

Cardiac alternans arises from dynamical instabilities in the electrical and calcium cycling systems of the heart, and often precedes ventricular arrhythmias and sudden cardiac death. In this review, we integrate clinical observations with theory and experiment to paint a holistic portrait of cardiac alternans: the underlying mechanisms, arrhythmic manifestations and electrocardiographic signatures. We first summarize the cellular and tissue mechanisms of alternans that have been demonstrated both theoretically and experimentally, including 3 voltage-driven and 2 calcium-driven alternans mechanisms. Based on experimental and simulation results, we describe their relevance to mechanisms of arrhythmogenesis under different disease conditions, and their link to electrocardiographic characteristics of alternans observed in patients. Our major conclusion is that alternans is not only a predictor, but also a causal mechanism of potentially lethal ventricular and atrial arrhythmias across the full spectrum of arrhythmia mechanisms that culminate in functional reentry, although less important for anatomic reentry and focal arrhythmias.

Activation of small conductance Ca2+ -activated K+ channels suppresses Ca2+ transient and action potential alternans in ventricular myocytes

At the cellular level, cardiac alternans is observed as beat-to-beat alternations in contraction strength, action potential (AP) morphology and Ca2+ transient (CaT) amplitude, and is a risk factor for cardiac arrhythmia. The (patho)physiological roles of small conductance Ca2+ -activated K+ (SK) channels in ventricles are poorly understood. We tested the hypothesis that in single rabbit ventricular myocytes pharmacological modulation of SK channels plays a causative role for the development of pacing-induced CaT and AP duration (APD) alternans. SK channel blockers (apamin, UCL1684) had only a minor effect on AP repolarization. However, SK channel activation by NS309 resulted in significant APD shortening, demonstrating that functional SK channels are well expressed in ventricular myocytes. The effects of NS309 were prevented or reversed by apamin and UCL1684, indicating that NS309 acted on SK channels. SK channel activation abolished or reduced the degree of pacing-induced CaT and APD alternans. Inhibition of KV 7.1 (with HMR1556) and KV 11.1 (with E4031) channels was used to mimic conditions of long QT syndromes type-1 and type-2, respectively. Both HMR1556 and E4031 enhanced CaT alternans that was prevented by SK channel activation. In AP voltage-clamped cells the SK channel activator had no effect on CaT alternans, confirming that suppression of CaT alternans was caused by APD shortening. APD shortening contributed to protection from alternans by lowering sarcoplasmic reticulum Ca2+ content and curtailing Ca2+ release. The data suggest that SK activation could be a potential intervention to avert development of alternans with important ramifications for arrhythmia prevention and therapy for patients with long QT syndrome. KEY POINTS: At the cellular level, cardiac alternans is observed as beat-to-beat alternations in contraction strength, action potential (AP) morphology and intracellular Ca2+ release amplitude, and is a risk factor for cardiac arrhythmia. The (patho)physiological roles of small conductance Ca2+ -activated K+ (SK) channels in ventricles are poorly understood. We investigated whether pharmacological modulation of SK channels affects the development of cardiac alternans in normal ventricular cells and in cells with drug-induced long QT syndrome (LQTS). While SK channel blockers have only a minor effect on AP morphology, their activation leads to AP shortening and abolishes or reduces the degree of pacing-induced Ca2+ and AP alternans. AP shortening contributed to protection against alternans by lowering sarcoplasmic reticulum Ca2+ content and curtailing Ca2+ release. The data suggest SK activation as a potential intervention to avert the development of alternans with important ramifications for arrhythmia prevention for patients with LQTS.

The 'Reverse FDUF' Mechanism of Atrial Excitation-Contraction Coupling Sustains Calcium Alternans-A Hypothesis

Cardiac calcium alternans is defined as beat-to-beat alternations of Ca transient (CaT) amplitude and has been linked to cardiac arrhythmia, including atrial fibrillation. We investigated the mechanism of atrial alternans in isolated rabbit atrial myocytes using high-resolution line scan confocal Ca imaging. Alternans was induced by increasing the pacing frequency until stable alternans was observed (1.6-2.5 Hz at room temperature). In atrial myocytes, action potential-induced Ca release is initiated in the cell periphery and subsequently propagates towards the cell center by Ca-induced Ca release (CICR) in a Ca wave-like fashion, driven by the newly identified 'fire-diffuse-uptake-fire' (FDUF) mechanism. The development of CaT alternans was accompanied by characteristic changes of the spatio-temporal organization of the CaT. During the later phase of the CaT, central [Ca]_i exceeded peripheral [Ca]_i that was indicative of a reversal of the subcellular [Ca]_i gradient from centripetal to centrifugal. This gradient reversal resulted in a reversal of CICR propagation, causing a secondary Ca release during the large-amplitude alternans CaT, thereby prolonging the CaT, enhancing Ca-release refractoriness and reducing Ca release on the subsequent beat, thus enhancing the degree of CaT alternans. Here, we propose the 'reverse FDUF' mechanism as a novel cellular mechanism of atrial CaT alternans, which explains how the uncoupling of central from peripheral Ca release leads to the reversal of propagating CICR and to alternans.

Mechanism of carvedilol induced action potential and calcium alternans

Carvedilol is a nonspecific β-blocker clinically used for the treatment of cardiovascular diseases but has also been shown to have profound effects on excitation-contraction coupling and Ca signaling at the cellular level. We investigate the mechanism by which carvedilol facilitates Ca transient (CaT) and action potential duration (APD) alternans in rabbit atrial myocytes. Carvedilol lowered the frequency threshold for pacing-induced CaT alternans and facilitated alternans in a concentration-dependent manner. Carvedilol prolonged the sarcoplasmic reticulum (SR) Ca release refractoriness by significantly increasing the time constant τ of recovery of SR Ca release; however, no changes in L-type calcium current recovery from inactivation or SR Ca load were found after carvedilol treatment. Carvedilol enhanced the degree of APD alternans nearly two-fold. Carvedilol slowed the APD restitution kinetics and steepened the APD restitution curve at the pacing frequency (2 Hz) where alternans were elicited. No effect on the CaT or APD alternans ratios was observed in experiments with a different β-blocker (metoprolol), excluding the possibility that the carvedilol effect on CaT and APD alternans was determined by its β-blocking properties. These data suggest that carvedilol contributes to the generation of CaT and APD alternans in atrial myocytes by modulating the restitution of CaT and APD.

Synchronization of spatially discordant voltage and calcium alternans in cardiac tissue

The heart is an excitable medium which is excited by membrane potential depolarization and propagation. Membrane potential depolarization brings in calcium (Ca) through the Ca channels to trigger intracellular Ca release for contraction of the heart. Ca also affects voltage via Ca-dependent ionic currents, and thus, voltage and Ca are bidirectionally coupled. It has been shown that the voltage subsystem or the Ca subsystem can generate its own dynamical instabilities which are affected by their bidirectional couplings, leading to complex dynamics of action potential and Ca cycling. Moreover, the dynamics become spatiotemporal in tissue in which cells are diffusively coupled through voltage. A widely investigated spatiotemporal dynamics is spatially discordant alternans (SDA) in which action potential duration (APD) or Ca amplitude exhibits temporally period-2 and spatially out-of-phase patterns, i.e., APD-SDA and Ca-SDA patterns, respectively. However, the mechanisms of formation, stability, and synchronization of APD-SDA and Ca-SDA patterns remain incompletely understood. In this paper, we use cardiac tissue models described by an amplitude equation, coupled iterated maps, and reaction-diffusion equations with detailed physiology (the ionic model) to perform analytical and computational investigations. We show that, when the Ca subsystem is stable, the Ca-SDA pattern always follows the APD-SDA pattern, and thus, they are always synchronized. When the Ca subsystem is unstable, synchronization of APD-SDA and Ca-SDA patterns depends on the stabilities of both subsystems, their coupling strengths, and the spatial scales of the initial Ca-SDA patterns. Spontaneous (initial condition-independent) synchronization is promoted by enhancing APD instability and reducing Ca instability as well as stronger Ca-to-APD and APD-to-Ca coupling, a pattern formation caused by dynamical instabilities. When Ca is more unstable and APD is less unstable or APD-to-Ca coupling is weak, synchronization of APD-SDA and Ca-SDA patterns is promoted by larger initially synchronized Ca-SDA clusters, i.e., initial condition-dependent synchronization. The synchronized APD-SDA and Ca-SDA patterns can be locked in-phase, antiphase, or quasiperiodic depending on the coupling relationship between APD and Ca. These theoretical and simulation results provide mechanistic insights into the APD-SDA and Ca-SDA dynamics observed in experimental studies.

An ensemble of parameters from a robust Markov-based model reproduces L-type calcium currents from different human cardiac myocytes

The development of modeling structures at the channel level that can integrate subcellular and cell models and properly reproduce different experimental data is of utmost importance in cardiac electrophysiology. In contrast to gate-based models, Markov Chain models are well suited to promote the integration of the subcellular level of the cardiomyocyte to the whole cell. In this paper, we develop Markov Chain models for the L-type Calcium current that can reproduce the electrophysiology of two established human models for the ventricular and Purkinje cells. In addition, instead of presenting a single set of parameters, we present a collection of set of parameters employing Differential Evolution algorithms that can properly reproduce very different protocol data. We show the importance of using an ensemble of a set of parameter values to obtain proper results when considering a second protocol that suppresses calcium inactivation and mimics a pathological condition. We discuss how model discrepancy, data availability, and parameter identifiability can influence the choice of the size of the collection. In summary, we have modified two cardiac models by proposing new Markov Chain models for the L-type Calcium. We keep the original whole-cell dynamics by reproducing the same characteristic action potential and calcium dynamics, whereas the Markov chain-based description of the L-type Calcium channels allows novel small spatial scale simulations of subcellular processes. Finally, the use of collections of parameters was crucial for addressing model discrepancy, identifiability issues, and avoiding fitting parameters overly precisely, i.e., overfitting.

Cardiac Alternans Occurs through the Synergy of Voltage- and Calcium-Dependent Mechanisms

Cardiac alternans is characterized by alternating weak and strong beats of the heart. This signaling at the cellular level may appear as alternating long and short action potentials (APs) that occur in synchrony with alternating large and small calcium transients, respectively. Previous studies have suggested that alternans manifests itself through either a voltage dependent mechanism based upon action potential restitution or as a calcium dependent mechanism based on refractoriness of calcium release. We use a novel model of cardiac excitation-contraction (EC) coupling in the rat ventricular myocyte that includes 20,000 calcium release units (CRU) each with 49 ryanodine receptors (RyR2s) and 7 L-type calcium channels that are all stochastically gated. The model suggests that at the cellular level in the case of alternans produced by rapid pacing, the mechanism requires a synergy of voltage- and calcium-dependent mechanisms. The rapid pacing reduces AP duration and magnitude reducing the number of L-type calcium channels activating individual CRUs during each AP and thus increases the population of CRUs that can be recruited stochastically. Elevated myoplasmic and sarcoplasmic reticulum (SR) calcium, [Ca2+]myo and [Ca2+]SR respectively, increases ryanodine receptor open probability (Po) according to our model used in this simulation and this increased the probability of activating additional CRUs. A CRU that opens in one beat is less likely to open the subsequent beat due to refractoriness caused by incomplete refilling of the junctional sarcoplasmic reticulum (jSR). Furthermore, the model includes estimates of changes in Na+ fluxes and [Na+]i and thus provides insight into how changes in electrical activity, [Na+]i and sodium-calcium exchanger activity can modulate alternans. The model thus tracks critical elements that can account for rate-dependent changes in [Na+]i and [Ca2+]myo and how they contribute to the generation of Ca2+ signaling alternans in the heart.

Excitation-contraction coupling and calcium release in atrial muscle

In cardiac muscle, the process of excitation-contraction coupling (ECC) describes the chain of events that links action potential induced myocyte membrane depolarization, surface membrane ion channel activation, triggering of Ca2+ induced Ca2+ release from the sarcoplasmic reticulum (SR) Ca2+ store to activation of the contractile machinery that is ultimately responsible for the pump function of the heart. Here we review similarities and differences of structural and functional attributes of ECC between atrial and ventricular tissue. We explore a novel "fire-diffuse-uptake-fire" paradigm of atrial ECC and Ca2+ release that assigns a novel role to the SR SERCA pump and involves a concerted "tandem" activation of the ryanodine receptor Ca2+ release channel by cytosolic and luminal Ca2+. We discuss the contribution of the inositol 1,4,5-trisphosphate (IP3) receptor Ca2+ release channel as an auxiliary pathway to Ca2+ signaling, and we review IP3 receptor-induced Ca2+ release involvement in beat-to-beat ECC, nuclear Ca2+ signaling, and arrhythmogenesis. Finally, we explore the topic of electromechanical and Ca2+ alternans and its ramifications for atrial arrhythmia.

Mitochondrial depolarization promotes calcium alternans: Mechanistic insights from a ventricular myocyte model

Mitochondria are vital organelles inside the cell and contribute to intracellular calcium (Ca2+) dynamics directly and indirectly via calcium exchange, ATP generation, and production of reactive oxygen species (ROS). Arrhythmogenic Ca2+ alternans in cardiac myocytes has been observed in experiments under abnormal mitochondrial depolarization. However, complex signaling pathways and Ca2+ cycling between mitochondria and cytosol make it difficult in experiments to reveal the underlying mechanisms of Ca2+ alternans under abnormal mitochondrial depolarization. In this study, we use a newly developed spatiotemporal ventricular myocyte computer model that integrates mitochondrial Ca2+ cycling and complex signaling pathways to investigate the mechanisms of Ca2+ alternans during mitochondrial depolarization. We find that elevation of ROS in response to mitochondrial depolarization plays a critical role in promoting Ca2+ alternans. Further examination reveals that the redox effect of ROS on ryanodine receptors and sarco/endoplasmic reticulum Ca2+-ATPase synergistically promote alternans. Upregulation of mitochondrial Ca2+ uniporter promotes Ca2+ alternans via Ca2+-dependent mitochondrial permeability transition pore opening. Due to their relatively slow kinetics, oxidized Ca2+/calmodulin-dependent protein kinase II activation and ATP do not play significant roles acutely in the genesis of Ca2+ alternans after mitochondrial depolarization, but their roles can be significant in the long term, mainly through their effects on sarco/endoplasmic reticulum Ca2+-ATPase activity. In conclusion, mitochondrial depolarization promotes Ca2+ alternans acutely via the redox effect of ROS and chronically by ATP reduction. It suppresses Ca2+ alternans chronically through oxidized Ca2+/calmodulin-dependent protein kinase II activation.

Heart failure-induced atrial remodelling promotes electrical and conduction alternans

Heart failure (HF) is associated with an increased propensity for atrial fibrillation (AF), causing higher mortality than AF or HF alone. It is hypothesized that HF-induced remodelling of atrial cellular and tissue properties promotes the genesis of atrial action potential (AP) alternans and conduction alternans that perpetuate AF. However, the mechanism underlying the increased susceptibility to atrial alternans in HF remains incompletely elucidated. In this study, we investigated the effects of how HF-induced atrial cellular electrophysiological (with prolonged AP duration) and tissue structural (reduced cell-to-cell coupling caused by atrial fibrosis) remodelling can have an effect on the generation of atrial AP alternans and their conduction at the cellular and one-dimensional (1D) tissue levels. Simulation results showed that HF-induced atrial electrical remodelling prolonged AP duration, which was accompanied by an increased sarcoplasmic reticulum (SR) Ca2+ content and Ca2+ transient amplitude. Further analysis demonstrated that HF-induced atrial electrical remodelling increased susceptibility to atrial alternans mainly due to the increased sarcoplasmic reticulum Ca2+-ATPase (SERCA) Ca2+ reuptake, modulated by increased phospholamban (PLB) phosphorylation, and the decreased transient outward K+ current (Ito). The underlying mechanism has been suggested that the increased SR Ca2+ content and prolonged AP did not fully recover to their previous levels at the end of diastole, resulting in a smaller SR Ca2+ release and AP in the next beat. These produced Ca2+ transient alternans and AP alternans, and further caused AP alternans and Ca2+ transient alternans through Ca2+→AP coupling and AP→Ca2+ coupling, respectively. Simulation of a 1D tissue model showed that the combined action of HF-induced ion channel remodelling and a decrease in cell-to-cell coupling due to fibrosis increased the heart tissue's susceptibility to the formation of spatially discordant alternans, resulting in an increased functional AP propagation dispersion, which is pro-arrhythmic. These findings provide insights into how HF promotes atrial arrhythmia in association with atrial alternans.

Calcium Buffering in the Heart in Health and Disease

Changes of intracellular Ca²⁺ concentration regulate many aspects of cardiac myocyte function. About 99% of the cytoplasmic calcium in cardiac myocytes is bound to buffers, and their properties will therefore have a major influence on Ca²⁺ signaling. This article considers the fundamental properties and identities of the buffers and how to measure them. It reviews the effects of buffering on the systolic Ca²⁺ transient and how this may change physiologically, and in heart failure and both atrial and ventricular arrhythmias, as well. It is concluded that the consequences of this strong buffering may be more significant than currently appreciated, and a fuller understanding is needed for proper understanding of cardiac calcium cycling and contractility.

Complex patterns of subcellular cardiac alternans

Cardiac alternans, in which the membrane potential and the intracellular calcium concentration exhibit alternating durations and peak amplitudes at consecutive beats, constitute a precursor to fatal cardiac arrhythmia such as sudden cardiac death. A crucial question therefore concerns the onset of cardiac alternans. Typically, alternans are only reported when they are fully developed. Here, we present a modelling approach to explore recently discovered microscopic alternans, which represent one of the earliest manifestations of cardiac alternans. In this case, the regular periodic dynamics of the local intracellular calcium concentration is already unstable, while the whole-cell behaviour suggests a healthy cell state. In particular, we use our model to investigate the impact of calcium diffusion in both the cytosol and the sarcoplasmic reticulum on the formation of microscopic calcium alternans. We find that for dominant cytosolic coupling, calcium alternans emerge via the traditional period doubling bifurcation. In contrast, dominant luminal coupling leads to a novel route to calcium alternans through a saddle-node bifurcation at the network level. Combining semi-analytical and computational approaches, we compute areas of stability in parameter space and find that as we cross from stable to unstable regions, the emergent patterns of the intracellular calcium concentration change abruptly in a fashion that is highly dependent upon position along the stability boundary. Our results demonstrate that microscopic calcium alternans may possess a much richer dynamical repertoire than previously thought and further strengthen the role of luminal calcium in shaping cardiac calcium dynamics.

Altered Intracellular Calcium Homeostasis and Arrhythmogenesis in the Aged Heart

Aging of the heart is associated with a blunted response to sympathetic stimulation, reduced contractility, and increased propensity for arrhythmias, with the risk of sudden cardiac death significantly increased in the elderly population. The altered cardiac structural and functional phenotype, as well as age-associated prevalent comorbidities including hypertension and atherosclerosis, predispose the heart to atrial fibrillation, heart failure, and ventricular tachyarrhythmias. At the cellular level, perturbations in mitochondrial function, excitation-contraction coupling, and calcium homeostasis contribute to this electrical and contractile dysfunction. Major determinants of cardiac contractility are the intracellular release of Ca²⁺ from the sarcoplasmic reticulum by the ryanodine receptors (RyR2), and the following sequestration of Ca²⁺ by the sarco/endoplasmic Ca²⁺-ATPase (SERCa2a). Activity of RyR2 and SERCa2a in myocytes is not only dependent on expression levels and interacting accessory proteins, but on fine-tuned regulation via post-translational modifications. In this paper, we review how aberrant changes in intracellular Ca²⁺ cycling via these proteins contributes to arrhythmogenesis in the aged heart.

Stochastic coupled map model of subcellular calcium cycling in cardiac cells

In this study, we analyze a nonlinear map model of intracellular calcium (Ca) and voltage in cardiac cells. In this model, Ca release from the sarcoplasmic reticulum (SR) occurs at spatially distributed dyadic junctions that are diffusively coupled. At these junctions, release occurs with a probability that depends on key variables such as the SR load and the diastolic interval. Using this model, we explore how nonlinearity and stochasticity determine the spatial distribution of Ca release events within a cardiac cell. In particular, we identify a novel synchronization transition, which occurs at rapid pacing rates, in which the global Ca transient transitions from a period 2 response to a period 1 response. In the global period 2 response dyadic junctions fire in unison, on average, on alternate beats, while in the period 1 regime, Ca release at individual dyads is highly irregular. A close examination of the spatial distribution of Ca reveals that in the period 1 regime, the system coarsens into spatially out-of-phase regions with a length scale much smaller than the system size, but larger than the spacing between dyads. We have also explored in detail the coupling to membrane voltage. We study first the case of positive coupling, where a large Ca transient promotes a long action potential duration (APD). Here, the coupling to voltage synchronizes Ca release so that the system exhibits a robust period 2 response that is independent of initial conditions. On the other hand, in the case of negative coupling, where a large Ca transient tends to shorten the APD, we find a multitude of metastable states which consist of complex spatially discordant alternans patterns. Using an analogy to equilibrium statistical mechanics, we show that the spatial patterns observed can be explained by a mapping to the Potts model, with an additional term that accounts for a global coupling of spin states. Using this analogy, we argue that Ca cycling in cardiac cells exhibits complex spatiotemporal patterns that emerge via first or second order phase transitions. These results show that voltage and Ca can interact in order to induce complex subcellular responses, which can potentially lead to heart rhythm disorders.

Action potential shortening rescues atrial calcium alternans

KEY POINTS

Cardiac alternans refers to a beat-to-beat alternation in contraction, action potential (AP) morphology and Ca²⁺ transient (CaT) amplitude, and represents a risk factor for cardiac arrhythmia, including atrial fibrillation. We developed strategies to pharmacologically manipulate the AP waveform with the goal to reduce or eliminate the occurrence of CaT and contraction alternans in atrial tissue. With combined patch-clamp and intracellular Ca²⁺ measurements we investigated the effect of specific ion channel inhibitors and activators on alternans. In single rabbit atrial myocytes, suppression of Ca²⁺ -activated Cl^- channels eliminated AP duration alternans, but prolonged the AP and failed to eliminate CaT alternans. In contrast, activation of K⁺ currents (I_Ks and I_Kr ) shortened the AP and eliminated both AP duration and CaT alternans. As demonstrated also at the whole heart level, activation of K⁺ conductances represents a promising strategy to suppress alternans, and thus reducing a risk factor for atrial fibrillation.

ABSTRACT

At the cellular level alternans is observed as beat-to-beat alternations in contraction, action potential (AP) morphology and magnitude of the Ca²⁺ transient (CaT). Alternans is a well-established risk factor for cardiac arrhythmia, including atrial fibrillation. This study investigates whether pharmacological manipulation of AP morphology is a viable strategy to reduce the risk of arrhythmogenic CaT alternans. Pacing-induced AP and CaT alternans were studied in rabbit atrial myocytes using combined Ca²⁺ imaging and electrophysiological measurements. Increased AP duration (APD) and beat-to-beat alternations in AP morphology lowered the pacing frequency threshold and increased the degree of CaT alternans. Inhibition of Ca²⁺ -activated Cl^- channels reduced beat-to-beat AP alternations, but prolonged APD and failed to suppress CaT alternans. In contrast, AP shortening induced by activators of two K⁺ channels (ML277 for Kv7.1 and NS1643 for Kv11.1) abolished both APD and CaT alternans in field-stimulated and current-clamped myocytes. K⁺ channel activators had no effect on the degree of Ca²⁺ alternans in AP voltage-clamped cells, confirming that suppression of Ca²⁺ alternans was caused by the changes in AP morphology. Finally, activation of Kv11.1 channel significantly attenuated or even abolished atrial T-wave alternans in isolated Langendorff perfused hearts. In summary, AP shortening suppressed or completely eliminated both CaT and APD alternans in single atrial myocytes and atrial T-wave alternans at the whole heart level. Therefore, we suggest that AP shortening is a potential intervention to avert development of alternans with important ramifications for arrhythmia prevention and therapy.

Proarrhythmic Remodeling of Calcium Homeostasis in Cardiac Disease; Implications for Diabetes and Obesity

A rapid growth in the incidence of diabetes and obesity has transpired to a major heath issue and economic burden in the postindustrial world, with more than 29 million patients affected in the United States alone. Cardiovascular defects have been established as the leading cause of mortality and morbidity of diabetic patients. Over the last decade, significant progress has been made in delineating mechanisms responsible for the diminished cardiac contractile function and enhanced propensity for malignant cardiac arrhythmias characteristic of diabetic disease. Rhythmic cardiac contractility relies upon the precise interplay between several cellular Ca²⁺ transport protein complexes including plasmalemmal L-type Ca²⁺ channels (LTCC), Na⁺-Ca²⁺ exchanger (NCX1), Sarco/endoplasmic Reticulum (SR) Ca²⁺-ATPase (SERCa2a) and ryanodine receptors (RyR2s), the SR Ca²⁺ release channels. Here we provide an overview of changes in Ca²⁺ homeostasis in diabetic ventricular myocytes and discuss the therapeutic potential of targeting Ca²⁺ handling proteins in the prevention of diabetes-associated cardiomyopathy and arrhythmogenesis.

Proarrhythmic Remodeling of Calcium Homeostasis in Cardiac Disease; Implications for Diabetes and Obesity

A rapid growth in the incidence of diabetes and obesity has transpired to a major heath issue and economic burden in the postindustrial world, with more than 29 million patients affected in the United States alone. Cardiovascular defects have been established as the leading cause of mortality and morbidity of diabetic patients. Over the last decade, significant progress has been made in delineating mechanisms responsible for the diminished cardiac contractile function and enhanced propensity for malignant cardiac arrhythmias characteristic of diabetic disease. Rhythmic cardiac contractility relies upon the precise interplay between several cellular Ca²⁺ transport protein complexes including plasmalemmal L-type Ca²⁺ channels (LTCC), Na⁺-Ca²⁺ exchanger (NCX1), Sarco/endoplasmic Reticulum (SR) Ca²⁺-ATPase (SERCa2a) and ryanodine receptors (RyR2s), the SR Ca²⁺ release channels. Here we provide an overview of changes in Ca²⁺ homeostasis in diabetic ventricular myocytes and discuss the therapeutic potential of targeting Ca²⁺ handling proteins in the prevention of diabetes-associated cardiomyopathy and arrhythmogenesis.

Proarrhythmic Remodeling of Calcium Homeostasis in Cardiac Disease; Implications for Diabetes and Obesity

A rapid growth in the incidence of diabetes and obesity has transpired to a major heath issue and economic burden in the postindustrial world, with more than 29 million patients affected in the United States alone. Cardiovascular defects have been established as the leading cause of mortality and morbidity of diabetic patients. Over the last decade, significant progress has been made in delineating mechanisms responsible for the diminished cardiac contractile function and enhanced propensity for malignant cardiac arrhythmias characteristic of diabetic disease. Rhythmic cardiac contractility relies upon the precise interplay between several cellular Ca²⁺ transport protein complexes including plasmalemmal L-type Ca²⁺ channels (LTCC), Na⁺-Ca²⁺ exchanger (NCX1), Sarco/endoplasmic Reticulum (SR) Ca²⁺-ATPase (SERCa2a) and ryanodine receptors (RyR2s), the SR Ca²⁺ release channels. Here we provide an overview of changes in Ca²⁺ homeostasis in diabetic ventricular myocytes and discuss the therapeutic potential of targeting Ca²⁺ handling proteins in the prevention of diabetes-associated cardiomyopathy and arrhythmogenesis.

Transverse tubular network structures in the genesis of intracellular calcium alternans and triggered activity in cardiac cells

RATIONALE

The major role of a transverse-tubular (TT) network in a cardiac cell is to facilitate effective excitation-contraction coupling and signaling. The TT network structures are heterogeneous within a single cell, and vary between different types of cells and species. They are also remodeled in cardiac diseases. However, how different TT network structures predispose cardiac cells to arrhythmogenesis remains to be revealed.

OBJECTIVE

To systematically investigate the roles of TT network structure and the underlying mechanisms in the genesis of intracellular calcium (Ca2+) alternans and triggered activity (TA).

METHODS AND RESULTS

Based on recent experimental observations, different TT network structures, including uniformly and non-uniformly random TT distributions, were modeled in a cardiac cell model consisting of a three-dimensional network of Ca2+ release units (CRUs). Our simulations showed that both Ca2+ alternans and Ca2+ wave-mediated TA were promoted when the fraction of orphaned CRUs was in an intermediate range, but suppressed in cells exhibiting either well-organized TT networks or low TT densities. Ca2+ alternans and TA could be promoted by low TT densities when the cells were small or the CRU coupling was strong. Both alternans and TA occurred more easily in uniformly random TT networks than in non-uniformly random TT networks. Subcellular spatially discordant Ca2+ alternans was promoted by non-uniformly random TT networks but suppressed by increasing CRU coupling strength. These mechanistic insights provide a holistic understanding of the effects of TT network structure on the susceptibility to arrhythmogenesis.

CONCLUSIONS

The TT network plays important roles in promoting Ca2+ alternans and TA, and different TT network structures may predispose cardiac cells differently to arrhythmogenesis.

Minimal model for calcium alternans due to SR release refractoriness

In the heart, rapid pacing rates may induce alternations in the strength of cardiac contraction, termed pulsus alternans. Often, this is due to an instability in the dynamics of the intracellular calcium concentration, whose transients become larger and smaller at consecutive beats. This alternation has been linked experimentally and theoretically to two different mechanisms: an instability due to (1) a strong dependence of calcium release on sarcoplasmic reticulum (SR) load, together with a slow calcium reuptake into the SR or (2) to SR release refractoriness, due to a slow recovery of the ryanodine receptors (RyR2) from inactivation. The relationship between calcium alternans and refractoriness of the RyR2 has been more elusive than the corresponding SR Ca load mechanism. To study the former, we reduce a general calcium model, which mimics the deterministic evolution of a calcium release unit, to its most basic elements. We show that calcium alternans can be understood using a simple nonlinear equation for calcium concentration at the dyadic space, coupled to a relaxation equation for the number of recovered RyR2s. Depending on the number of RyR2s that are recovered at the beginning of a stimulation, the increase in calcium concentration may pass, or not, over an excitability threshold that limits the occurrence of a large calcium transient. When the recovery of the RyR2 is slow, this produces naturally a period doubling bifurcation, resulting in calcium alternans. We then study the effects of inactivation, calcium diffusion, and release conductance for the onset of alternans. We find that the development of alternans requires a well-defined value of diffusion while it is less sensitive to the values of inactivation or release conductance.

Studying dyadic structure-function relationships: a review of current modeling approaches and new insights into Ca2+ (mis)handling

Excitation–contraction coupling in cardiac myocytes requires calcium influx through L-type calcium channels in the sarcolemma, which gates calcium release through sarcoplasmic reticulum ryanodine receptors in a process known as calcium-induced calcium release, producing a myoplasmic calcium transient and enabling cardiomyocyte contraction. The spatio-temporal dynamics of calcium release, buffering, and reuptake into the sarcoplasmic reticulum play a central role in excitation–contraction coupling in both normal and diseased cardiac myocytes. However, further quantitative understanding of these cells’ calcium machinery and the study of mechanisms that underlie both normal cardiac function and calcium-dependent etiologies in heart disease requires accurate knowledge of cardiac ultrastructure, protein distribution and subcellular function. As current imaging techniques are limited in spatial resolution, limiting insight into changes in calcium handling, computational models of excitation–contraction coupling have been increasingly employed to probe these structure–function relationships. This review will focus on the development of structural models of cardiac calcium dynamics at the subcellular level, orienting the reader broadly towards the development of models of subcellular calcium handling in cardiomyocytes. Specific focus will be given to progress in recent years in terms of multi-scale modeling employing resolved spatial models of subcellular calcium machinery. A review of the state-of-the-art will be followed by a review of emergent insights into calcium-dependent etiologies in heart disease and, finally, we will offer a perspective on future directions for related computational modeling and simulation efforts.

Membrane potential determines calcium alternans through modulation of SR Ca2+ load and L-type Ca2+ current

Alternans is a risk factor for cardiac arrhythmia, including atrial fibrillation. At the cellular level alternans is observed as beat-to-beat alternations in contraction, action potential (AP) morphology and magnitude of the Ca²⁺ transient (CaT). It is widely accepted that the bi-directional interplay between membrane voltage and Ca²⁺ is crucial for the development of alternans, however recently the attention has shifted to instabilities in cellular Ca²⁺ handling, while the role of AP alternation remains poorly understood. This study provides new insights how beat- to-beat alternation in AP morphology affects occurrence of CaT alternans in atrial myocytes. Pacing-induced AP and CaT alternans were studied in rabbit atrial myocytes using combined Ca²⁺ imaging and electrophysiological measurements. To determine the role of AP morphology for the generation of CaT alternans, trains of two voltage commands in form of APs recorded during large and small alternans CaTs were applied to voltage-clamped cells. APs of longer duration (as observed during small amplitude alternans CaT) and especially beat-to-beat alternations in AP morphology (AP alternans) reduced the pacing frequency threshold and increased the degree of CaT alternans. AP morphology contributes to the development of CaT alternans by two mechanisms. First, the AP waveform observed during small alternans CaTs coincided with higher end-diastolic sarcoplasmic reticulum Ca²⁺ levels ([Ca²⁺]_SR), and AP alternans resulted in beat-to-beat alternations in end-diastolic [Ca²⁺]_SR. Second, L-type Ca²⁺ current was significantly affected by AP morphology, where the AP waveform observed during large CaT elicited L-type Ca²⁺ currents of higher magnitude and faster kinetics, resulting in more efficient triggering of SR Ca²⁺ release. In conclusion, alternation in AP morphology plays a significant role in the development and stabilization of atrial alternans. The demonstration that CaT alternans can be controlled or even prevented by modulating AP morphology has important ramifications for arrhythmia prevention and therapy strategies.

A unified theory of calcium alternans in ventricular myocytes

Intracellular calcium (Ca²⁺) alternans is a dynamical phenomenon in ventricular myocytes, which is linked to the genesis of lethal arrhythmias. Iterated map models of intracellular Ca²⁺ cycling dynamics in ventricular myocytes under periodic pacing have been developed to study the mechanisms of Ca²⁺ alternans. Two mechanisms of Ca²⁺ alternans have been demonstrated in these models: one relies mainly on fractional sarcoplasmic reticulum Ca²⁺ release and uptake, and the other on refractoriness and other properties of Ca²⁺ sparks. Each of the two mechanisms can partially explain the experimental observations, but both have their inconsistencies with the experimental results. Here we developed an iterated map model that is composed of two coupled iterated maps, which unifies the two mechanisms into a single cohesive mathematical framework. The unified theory can consistently explain the seemingly contradictory experimental observations and shows that the two mechanisms work synergistically to promote Ca²⁺ alternans. Predictions of the theory were examined in a physiologically-detailed spatial Ca²⁺ cycling model of ventricular myocytes.

Ca(2+)-activated chloride channel activity during Ca(2+) alternans in ventricular myocytes

Cardiac alternans, defined beat-to-beat alternations in contraction, action potential (AP) morphology or cytosolic Ca transient (CaT) amplitude, is a high risk indicator for cardiac arrhythmias. We investigated mechanisms of cardiac alternans in single rabbit ventricular myocytes. CaTs were monitored simultaneously with membrane currents or APs recorded with the patch clamp technique. A strong correlation between beat-to-beat alternations of AP morphology and CaT alternans was observed. During CaT alternans application of voltage clamp protocols in form of pre-recorded APs revealed a prominent Ca(2+)-dependent membrane current consisting of a large outward component coinciding with AP phases 1 and 2, followed by an inward current during AP repolarization. Approximately 85% of the initial outward current was blocked by Cl(-) channel blocker DIDS or lowering external Cl(-) concentration identifying it as a Ca(2+)-activated Cl(-) current (ICaCC). The data suggest that ICaCC plays a critical role in shaping beat-to-beat alternations in AP morphology during alternans.

SERCA2a upregulation ameliorates cellular alternans induced by metabolic inhibition

Cardiac alternans has been associated with the incidence of ventricular tachyarrhythmias and sudden cardiac death. The aim of this study was to investigate the effect of impaired mitochondrial function in the genesis of cellular alternans and to examine whether modulating the sarcoplasmic reticulum (SR) Ca(2+)ameliorates the level of alternans. Cardiomyocytes isolated from control and doxycyline-induced sarco(endo)plasmic reticulum Ca(2+)-ATPase 2a (SERCA2a)-upregulated mice were loaded with two different Ca(2+)indicators to selectively measure mitochondrial and cytosolic Ca(2+)using a custom-made fluorescence photometry system. The degree of alternans was defined as the alternans ratio (AR) [1 - (small Ca(2+)intensity)/(large Ca(2+)intensity)]. Blocking of complex I and II, cytochrome-coxidase, F0F1synthase, α-ketoglutarate dehydrogenase of the electron transport chain, increased alternans in both control and SERCA2a mice (P< 0.01). Changes in AR in SERCA2a-upregulated mice were significantly less pronounced than those observed in control in seven of nine tested conditions (P< 0.04).N-acetyl-l-cysteine (NAC), rescued alternans in myocytes that were previously exposed to an oxidizing agent (P< 0.001). CGP, an antagonist of the mitochondrial Na(+)-Ca(2+)exchanger, had the most severe effect on AR. Exposure to cyclosporin A, a blocker of the mitochondrial permeability transition pore reduced CGP-induced alternans (P< 0.0001). The major findings of this study are that impairment of mitochondrial Ca(2+)cycling and energy production leads to a higher amplitude of alternans in both control and SERCA2a-upregulated mice, but changes in SERCA2a-upregulated mice are less severe, indicating that SERCA2a mice are more capable of sustaining electrical stability during stress. This suggests a relationship between sarcoplasmic Ca(2+)content and mitochondrial dysfunction during alternans, which may potentially help to understand changes in Ca(2+)signaling in myocytes from diseased hearts, leading to new therapeutic targets.

Calcium-activated chloride current determines action potential morphology during calcium alternans in atrial myocytes

KEY POINTS

Cardiac alternans--periodic beat-to-beat alternations in contraction, action potential (AP) morphology or cytosolic calcium transient (CaT) amplitude--is a high risk indicator for cardiac arrhythmias and sudden cardiac death. However, it remains an unresolved issue whether beat-to-beat alternations in intracellular Ca(2+) ([Ca(2+)]i ) or AP morphology are the primary cause of pro-arrhythmic alternans. Here we show that in atria AP alternans occurs secondary to CaT alternans. CaT alternans leads to complex beat-to-beat changes in Ca(2+)-regulated ion currents that determine alternans of AP morphology. We report the novel finding that alternans of AP morphology is largely sustained by the activity of Ca(2+)-activated Cl(-) channels (CaCCs). Suppression of the CaCCs significantly reduces AP alternans, while CaT alternans remains unaffected. The demonstration of a major role of CaCCs in the development of AP alternans opens new possibilities for atrial alternans and arrhythmia prevention. Cardiac alternans, described as periodic beat-to-beat alternations in contraction, action potential (AP) morphology or cytosolic Ca transient (CaT) amplitude, is a high risk indicator for cardiac arrhythmias and sudden cardiac death. We investigated mechanisms of cardiac alternans in single rabbit atrial myocytes. CaTs were monitored simultaneously with membrane currents or APs recorded with the patch clamp technique. Beat-to-beat alternations of AP morphology and CaT amplitude revealed a strong quantitative correlation. Application of voltage clamp protocols in the form of pre-recorded APs (AP-clamp) during pacing-induced CaT alternans revealed a Ca(2+)-dependent current consisting of a large outward component (4.78 ± 0.58 pA pF(-1) in amplitude) coinciding with AP phases 1 and 2 that was followed by an inward current (-0.42 ± 0.03 pA pF(-1); n = 21) during AP repolarization. Approximately 90% of the initial outward current was blocked by substitution of Cl(-) ions or application of the Cl(-) channel blocker DIDS identifying it as a Ca(2+)-activated Cl(-) current (ICaCC). The prominent AP prolongation at action potential duration at 30% repolarization level during the small alternans CaT was due to reduced ICaCC. Inhibition of Cl(-) currents abolished AP alternans, but failed to affect CaT alternans, indicating that disturbances in Ca(2+) signalling were the primary event leading to alternans, and ICaCC played a decisive role in shaping the beat-to-beat alternations in AP morphology observed during alternans.

Cardiac alternans and intracellular calcium cycling

Cardiac alternans refers to a condition in which there is a periodic beat-to-beat oscillation in electrical activity and the strength of cardiac muscle contraction at a constant heart rate. Clinically, cardiac alternans occurs in settings that are typical for cardiac arrhythmias and has been causally linked to these conditions. At the cellular level, alternans is defined as beat-to-beat alternations in contraction amplitude (mechanical alternans), action potential duration (APD; electrical or APD alternans) and Ca(2+) transient amplitude (Ca(2+) alternans). The cause of alternans is multifactorial; however, alternans always originate from disturbances of the bidirectional coupling between membrane voltage (Vm ) and intracellular calcium ([Ca(2+) ]i ). Bidirectional coupling refers to the fact that, in cardiac cells, Vm depolarization and the generation of action potentials cause the elevation of [Ca(2+) ]i that is required for contraction (a process referred to as excitation-contraction coupling); conversely, changes of [Ca(2+) ]i control Vm because important membrane currents are Ca(2+) dependent. Evidence is mounting that alternans is ultimately caused by disturbances of cellular Ca(2+) signalling. Herein we review how two key factors of cardiac cellular Ca(2+) cycling, namely the release of Ca(2+) from internal stores and the capability of clearing the cytosol from Ca(2+) after each beat, determine the conditions under which alternans occurs. The contributions from key Ca(2+) -handling proteins (i.e. surface membrane channels, ion pumps and transporters and internal Ca(2+) release channels) are discussed.

Voltage and calcium dynamics both underlie cellular alternans in cardiac myocytes

Cardiac alternans, a putative trigger event for cardiac reentry, is a beat-to-beat alternation in membrane potential and calcium transient. Alternans was originally attributed to instabilities in transmembrane ion channel dynamics (i.e., the voltage mechanism). As of this writing, the predominant view is that instabilities in subcellular calcium handling are the main underlying mechanism. That being said, because the voltage and calcium systems are bidirectionally coupled, theoretical studies have suggested that both mechanisms can contribute. To date, to our knowledge, no experimental evidence of such a dual role within the same cell has been reported. Here, a combined electrophysiological and calcium imaging approach was developed and used to illuminate the contributions of voltage and calcium dynamics to alternans. An experimentally feasible protocol, quantification of subcellular calcium alternans and restitution slope during cycle-length ramping alternans control, was designed and validated. This approach allows simultaneous illumination of the contributions of voltage and calcium-driven instability to total cellular instability as a function of cycle-length. Application of this protocol in in vitro guinea-pig left-ventricular myocytes demonstrated that both voltage- and calcium-driven instabilities underlie alternans, and that the relative contributions of the two systems change as a function of pacing rate.

The mechanisms of calcium cycling and action potential dynamics in cardiac alternans

RATIONALE

Alternans is a risk factor for cardiac arrhythmia, including atrial fibrillation. At the cellular level alternans manifests as beat-to-beat alternations in contraction, action potential duration (APD), and magnitude of the Ca(2+) transient (CaT). Electromechanical and CaT alternans are highly correlated, however, it has remained controversial whether the primary cause of alternans is a disturbance of cellular Ca(2+) signaling or electrical membrane properties.

OBJECTIVE

To determine whether a primary failure of intracellular Ca(2+) regulation or disturbances in membrane potential and AP regulation are responsible for the occurrence of alternans in atrial myocytes.

METHODS AND RESULTS

Pacing-induced APD and CaT alternans were studied in single rabbit atrial and ventricular myocytes using combined [Ca(2+)]i and electrophysiological measurements. In current-clamp experiments, APD and CaT alternans strongly correlated in time and magnitude. CaT alternans was observed without alternation in L-type Ca(2+) current, however, elimination of intracellular Ca(2+) release abolished APD alternans, indicating that [Ca(2+)]i dynamics have a profound effect on the occurrence of CaT alternans. Trains of 2 distinctive voltage commands in form of APs recorded during large and small alternans CaTs were applied to voltage-clamped cells. CaT alternans was observed with and without alternation in the voltage command shape. During alternans AP-clamp large CaTs coincided with both long and short AP waveforms, indicating that CaT alternans develop irrespective of AP dynamics.

CONCLUSIONS

The primary mechanism underlying alternans in atrial cells, similarly to ventricular cells, resides in a disturbance of Ca(2+) signaling, whereas APD alternans are a secondary consequence, mediated by Ca(2+)-dependent AP modulation.

T-tubule disruption promotes calcium alternans in failing ventricular myocytes: mechanistic insights from computational modeling

In heart failure (HF), T-tubule (TT) disruption contributes to dyssynchronous calcium (Ca) release and impaired contraction, but its role in arrhythmogenesis remains unclear. In this study, we investigate the effects of TT disruption and other HF remodeling factors on Ca alternans in ventricular myocytes using computer modeling. A ventricular myocyte model with detailed spatiotemporal Ca cycling modeled by a coupled Ca release unit (CRU) network was used, in which the L-type Ca channels and the ryanodine receptor (RyR) channels were simulated by random Markov transitions. TT disruption, which removes the L-type Ca channels from the associated CRUs, results in "orphaned" RyR clusters and thus provides increased opportunity for spark-induced Ca sparks to occur. This effect combined with other HF remodeling factors promoted alternans by two distinct mechanisms: 1) for normal sarco-endoplasmic reticulum Ca ATPase (SERCA) activity, alternans was caused by both CRU refractoriness and coupling. The increased opportunity for spark-induced sparks by TT disruption combined with the enhanced CRU coupling by Ca elevation in the presence or absence of increased RyR leakiness facilitated spark synchronization on alternate beats to promote Ca alternans; 2) for down-regulated SERCA, alternans was caused by the sarcoplasmic reticulum (SR) Ca load-dependent mechanism, independent of CRU refractoriness. TT disruption and increased RyR leakiness shifted and steepened the SR Ca release-load relationship, which combines with down-regulated SERCA to promote Ca alternans. In conclusion, the mechanisms of Ca alternans for normal and down-regulated SERCA are different, and TT disruption promotes Ca alternans by both mechanisms, which may contribute to alternans at different stages of HF.

Mechanisms of ventricular arrhythmias: from molecular fluctuations to electrical turbulence

Ventricular arrhythmias have complex causes and mechanisms. Despite extensive investigation involving many clinical, experimental, and computational studies, effective biological therapeutics are still very limited. In this article, we review our current understanding of the mechanisms of ventricular arrhythmias by summarizing the state of knowledge spanning from the molecular scale to electrical wave behavior at the tissue and organ scales and how the complex nonlinear interactions integrate into the dynamics of arrhythmias in the heart. We discuss the challenges that we face in synthesizing these dynamics to develop safe and effective novel therapeutic approaches.

Nonlinear and Stochastic Dynamics in the Heart

In a normal human life span, the heart beats about 2 to 3 billion times. Under diseased conditions, a heart may lose its normal rhythm and degenerate suddenly into much faster and irregular rhythms, called arrhythmias, which may lead to sudden death. The transition from a normal rhythm to an arrhythmia is a transition from regular electrical wave conduction to irregular or turbulent wave conduction in the heart, and thus this medical problem is also a problem of physics and mathematics. In the last century, clinical, experimental, and theoretical studies have shown that dynamical theories play fundamental roles in understanding the mechanisms of the genesis of the normal heart rhythm as well as lethal arrhythmias. In this article, we summarize in detail the nonlinear and stochastic dynamics occurring in the heart and their links to normal cardiac functions and arrhythmias, providing a holistic view through integrating dynamics from the molecular (microscopic) scale, to the organelle (mesoscopic) scale, to the cellular, tissue, and organ (macroscopic) scales. We discuss what existing problems and challenges are waiting to be solved and how multi-scale mathematical modeling and nonlinear dynamics may be helpful for solving these problems.

Mitochondrial calcium uptake

Calcium (Ca(2+)) uptake into the mitochondrial matrix is critically important to cellular function. As a regulator of matrix Ca(2+) levels, this flux influences energy production and can initiate cell death. If large, this flux could potentially alter intracellular Ca(2+) ([Ca(2+)]i) signals. Despite years of study, fundamental disagreements on the extent and speed of mitochondrial Ca(2+) uptake still exist. Here, we review and quantitatively analyze mitochondrial Ca(2+) uptake fluxes from different tissues and interpret the results with respect to the recently proposed mitochondrial Ca(2+) uniporter (MCU) candidate. This quantitative analysis yields four clear results: (i) under physiological conditions, Ca(2+) influx into the mitochondria via the MCU is small relative to other cytosolic Ca(2+) extrusion pathways; (ii) single MCU conductance is ∼6-7 pS (105 mM [Ca(2+)]), and MCU flux appears to be modulated by [Ca(2+)]i, suggesting Ca(2+) regulation of MCU open probability (P(O)); (iii) in the heart, two features are clear: the number of MCU channels per mitochondrion can be calculated, and MCU probability is low under normal conditions; and (iv) in skeletal muscle and liver cells, uptake per mitochondrion varies in magnitude but total uptake per cell still appears to be modest. Based on our analysis of available quantitative data, we conclude that although Ca(2+) critically regulates mitochondrial function, the mitochondria do not act as a significant dynamic buffer of cytosolic Ca(2+) under physiological conditions. Nevertheless, with prolonged (superphysiological) elevations of [Ca(2+)]i, mitochondrial Ca(2+) uptake can increase 10- to 1,000-fold and begin to shape [Ca(2+)]i dynamics.

Calcium alternans in cardiac myocytes: order from disorder

Calcium alternans is associated with T-wave alternans and pulsus alternans, harbingers of increased mortality in the setting of heart disease. Recent experimental, computational, and theoretical studies have led to new insights into the mechanisms of Ca alternans, specifically how disordered behaviors dominated by stochastic processes at the subcellular level become organized into ordered periodic behaviors. In this article, we summarize the recent progress in this area, outlining a holistic theoretical framework in which the complex effects of Ca cycling proteins on Ca alternans are linked to three key properties of the cardiac Ca cycling network: randomness, refractoriness, and recruitment. We also illustrate how this '3R theory' can reconcile many seemingly contradictory experimental observations.