Alternans of cardiac calcium cycling in a cluster of ryanodine receptors: a simulation study

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.

Modeling Calcium Cycling in the Heart: Progress, Pitfalls, and Challenges

Intracellular calcium (Ca) cycling in the heart plays key roles in excitation-contraction coupling and arrhythmogenesis. In cardiac myocytes, the Ca release channels, i.e., the ryanodine receptors (RyRs), are clustered in the sarcoplasmic reticulum membrane, forming Ca release units (CRUs). The RyRs in a CRU act collectively to give rise to discrete Ca release events, called Ca sparks. A cell contains hundreds to thousands of CRUs, diffusively coupled via Ca to form a CRU network. A rich spectrum of spatiotemporal Ca dynamics is observed in cardiac myocytes, including Ca sparks, spark clusters, mini-waves, persistent whole-cell waves, and oscillations. Models of different temporal and spatial scales have been developed to investigate these dynamics. Due to the complexities of the CRU network and the spatiotemporal Ca dynamics, it is challenging to model the Ca cycling dynamics in the cardiac system, particularly at the tissue sales. In this article, we review the progress of modeling of Ca cycling in cardiac systems from single RyRs to the tissue scale, the pros and cons of the current models and different modeling approaches, and the challenges to be tackled in the future.

Voltage-mediated mechanism for calcium wave synchronization and arrhythmogenesis in atrial tissue

A wide range of atrial arrythmias are caused by molecular defects in proteins that regulate calcium (Ca) cycling. In many cases, these defects promote the propagation of subcellular Ca waves in the cell, which can perturb the voltage time course and induce dangerous perturbations of the action potential (AP). However, subcellular Ca waves occur randomly in cells and, therefore, electrical coupling between cells substantially decreases their effect on the AP. In this study, we present evidence that Ca waves in atrial tissue can synchronize in-phase owing to an order-disorder phase transition. In particular, we show that, below a critical pacing rate, Ca waves are desynchronized and therefore do not induce substantial AP fluctuations in tissue. However, above this critical pacing rate, Ca waves gradually synchronize over millions of cells, which leads to a dramatic amplification of AP fluctuations. We exploit an underlying Ising symmetry of paced cardiac tissue to show that this transition exhibits universal properties common to a wide range of physical systems in nature. Finally, we show that in the heart, phase synchronization induces spatially out-of-phase AP duration alternans which drives wave break and reentry. These results suggest that cardiac tissue exhibits a phase transition that is required for subcellular Ca cycling defects to induce a life-threatening arrhythmia.

Optical Ultrastructure of Large Mammalian Hearts Recovers Discordant Alternans by In Silico Data Assimilation

Understanding and predicting the mechanisms promoting the onset and sustainability of cardiac arrhythmias represent a primary concern in the scientific and medical communities still today. Despite the long-lasting effort in clinical and physico-mathematical research, a critical aspect to be fully characterized and unveiled is represented by spatiotemporal alternans patterns of cardiac excitation. The identification of discordant alternans and higher-order alternating rhythms by advanced data analyses as well as their prediction by reliable mathematical models represents a major avenue of research for a broad and multidisciplinary scientific community. Current limitations concern two primary aspects: 1) robust and general-purpose feature extraction techniques and 2) in silico data assimilation within reliable and predictive mathematical models. Here, we address both aspects. At first, we extend our previous works on Fourier transformation imaging (FFI), applying the technique to whole-ventricle fluorescence optical mapping. Overall, we identify complex spatial patterns of voltage alternans and characterize higher-order rhythms by a frequency-series analysis. Then, we integrate the optical ultrastructure obtained by FFI analysis within a fine-tuned electrophysiological mathematical model of the cardiac action potential. We build up a novel data assimilation procedure demonstrating its reliability in reproducing complex alternans patterns in two-dimensional computational domains. Finally, we prove that the FFI approach applied to both experimental and simulated signals recovers the same information, thus closing the loop between the experiment, data analysis, and numerical simulations.

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.

Cx43 hemichannel microdomain signaling at the intercalated disc enhances cardiac excitability

Cx43, a major cardiac connexin, forms precursor hemichannels that accrue at the intercalated disc to assemble as gap junctions. While gap junctions are crucial for electrical conduction in the heart, little is known about the potential roles of hemichannels. Recent evidence suggests that inhibiting Cx43 hemichannel opening with Gap19 has antiarrhythmic effects. Here, we used multiple electrophysiology, imaging, and super-resolution techniques to understand and define the conditions underlying Cx43 hemichannel activation in ventricular cardiomyocytes, their contribution to diastolic Ca2+ release from the sarcoplasmic reticulum, and their impact on electrical stability. We showed that Cx43 hemichannels were activated during diastolic Ca2+ release in single ventricular cardiomyocytes and cardiomyocyte cell pairs from mice and pigs. This activation involved Cx43 hemichannel Ca2+ entry and coupling to Ca2+ release microdomains at the intercalated disc, resulting in enhanced Ca2+ dynamics. Hemichannel opening furthermore contributed to delayed afterdepolarizations and triggered action potentials. In single cardiomyocytes, cardiomyocyte cell pairs, and arterially perfused tissue wedges from failing human hearts, increased hemichannel activity contributed to electrical instability compared with nonfailing rejected donor hearts. We conclude that microdomain coupling between Cx43 hemichannels and Ca2+ release is a potentially novel, targetable mechanism of cardiac arrhythmogenesis in heart failure.

Electrophysiological Mechanisms Underlying T-Wave Alternans and Their Role in Arrhythmogenesis

T-wave alternans (TWA) reflects every-other-beat alterations in the morphology of the electrocardiogram ST segment or T wave in the setting of a constant heart rate, hence, in the absence of heart rate variability. It is believed to be associated with the dispersion of repolarization and has been used as a non-invasive marker for predicting the risk of malignant cardiac arrhythmias and sudden cardiac death as numerous studies have shown. This review aims to provide up-to-date review on both experimental and simulation studies in elucidating possible mechanisms underlying the genesis of TWA at the cellular level, as well as the genesis of spatially concordant/discordant alternans at the tissue level, and their transition to cardiac arrhythmia. Recent progress and future perspectives in antiarrhythmic therapies associated with TWA are also discussed.

Delayed global feedback in the genesis and stability of spatiotemporal excitation patterns in paced biological excitable media

Biological excitable media, such as cardiac or neural cells and tissue, exhibit memory in which a change in the present excitation may affect the behaviors in the next excitation. For example, a change in calcium (Ca²⁺) concentration in a cell in the present excitation may affect the Ca²⁺ dynamics in the next excitation via bi-directional coupling between voltage and Ca²⁺, forming a delayed feedback loop. Since the Ca²⁺ dynamics inside the excitable cells are spatiotemporal while the membrane voltage is a global signal, the feedback loop is then a delayed global feedback (DGF) loop. In this study, we investigate the roles of DGF in the genesis and stability of spatiotemporal excitation patterns in periodically-paced excitable media using mathematical models with different levels of complexity: a model composed of coupled FitzHugh-Nagumo units, a 3-dimensional physiologically-detailed ventricular myocyte model, and a coupled map lattice model. We investigate the dynamics of excitation patterns that are temporal period-2 (P2) and spatially concordant or discordant, such as subcellular concordant or discordant Ca²⁺alternans in cardiac myocytes or spatially concordant or discordant Ca²⁺ and repolarization alternans in cardiac tissue. Our modeling approach allows both computer simulations and rigorous analytical treatments, which lead to the following results and conclusions. When DGF is absent, concordant and discordant P2 patterns occur depending on initial conditions with the discordant P2 patterns being spatially random. When the DGF is negative, only concordant P2 patterns exist. When the DGF is positive, both concordant and discordant P2 patterns can occur. The discordant P2 patterns are still spatially random, but they satisfy that the global signal exhibits a temporal period-1 behavior. The theoretical analyses of the coupled map lattice model reveal the underlying instabilities and bifurcations for the genesis, selection, and stability of spatiotemporal excitation patterns.

Understanding the mechanisms of pattern formation in biological systems is of great importance. Here we investigate the dynamical mechanisms by which delayed global feedback affects excitation pattern formation and stability in periodically-paced biological excitable media, such as cardiac or neural cells and tissue. We focus on the formation and stability of the temporal period-2 and spatially in-phase and out-of-phase excitation patterns. Using models of different levels of complexity, we show that when the delayed global feedback is negative, only the spatially in-phase patterns are stable. When the feedback is positive, both spatially in-phase and out-of-phase patterns are stable, and the out-of-phase patterns are spatially random but satisfy that the global signals are temporal period-1 solutions.

Mechanistic insight into spontaneous transition from cellular alternans to arrhythmia-A simulation study

Cardiac electrical alternans (CEA), manifested as T-wave alternans in ECG, is a clinical biomarker for predicting cardiac arrhythmias and sudden death. However, the mechanism underlying the spontaneous transition from CEA to arrhythmias remains incompletely elucidated. In this study, multiscale rabbit ventricular models were used to study the transition and a potential role of INa in perpetuating such a transition. It was shown CEA evolved into either concordant or discordant action potential (AP) conduction alternans in a homogeneous one-dimensional tissue model, depending on tissue AP duration and conduction velocity (CV) restitution properties. Discordant alternans was able to cause conduction failure in the model, which was promoted by impaired sodium channel with either a reduced or increased channel current. In a two-dimensional homogeneous tissue model, a combined effect of rate- and curvature-dependent CV broke-up alternating wavefronts at localised points, facilitating a spontaneous transition from CEA to re-entry. Tissue inhomogeneity or anisotropy further promoted break-up of re-entry, leading to multiple wavelets. Similar observations have also been seen in human atrial cellular and tissue models. In conclusion, our results identify a mechanism by which CEA spontaneously evolves into re-entry without a requirement for premature ventricular complexes or pre-existing tissue heterogeneities, and demonstrated the important pro-arrhythmic role of impaired sodium channel activity. These findings are model-independent and have potential human relevance.

Mechanisms linking T-wave alternans to spontaneous initiation of ventricular arrhythmias in rabbit models of long QT syndrome

KEY POINTS

T-wave alternans (TWA) and T-wave lability (TWL) are precursors of ventricular arrhythmias in long QT syndrome; however, the mechanistic link remains to be clarified. Computer simulations show that action potential duration (APD) prolongation and slowed heart rates promote APD alternans and chaos, manifesting as TWA and TWL, respectively. Regional APD alternans and chaos can exacerbate pre-existing or induce de novo APD dispersion, which combines with enhanced I_Ca,L to result in premature ventricular complexes (PVCs) originating from the APD gradient region. These PVCs can directly degenerate into re-entrant arrhythmias without the need for an additional tissue substrate or further exacerbate the APD dispersion to cause spontaneous initiation of ventricular arrhythmias. Experiments conducted in transgenic long QT rabbits show that PVC alternans occurs at slow heart rates, preceding spontaneous intuition of ventricular arrhythmias.

ABSTRACT

T-wave alternans (TWA) and irregular beat-to-beat T-wave variability or T-wave lability (TWL), the ECG manifestations of action potential duration (APD) alternans and variability, are precursors of ventricular arrhythmias in long QT syndromes. TWA and TWL in patients tend to occur at normal heart rates and are usually potentiated by bradycardia. Whether or how TWA and TWL at normal or slow heart rates are causally linked to arrhythmogenesis remains unknown. In the present study, we used computer simulations and experiments of a transgenic rabbit model of long QT syndrome to investigate the underlying mechanisms. Computer simulations showed that APD prolongation and slowed heart rates caused early afterdepolarization-mediated APD alternans and chaos, manifesting as TWA and TWL, respectively. Regional APD alternans and chaos exacerbated pre-existing APD dispersion and, in addition, APD chaos could also induce APD dispersion de novo via chaos desynchronization. Increased APD dispersion, combined with substantially enhanced I_Ca,L , resulted in a tissue-scale dynamical instability that gave rise to the spontaneous occurrence of unidirectionally propagating premature ventricular complexes (PVCs) originating from the APD gradient region. These PVCs could directly degenerate into re-entrant arrhythmias without the need for an additional tissue substrate or could block the following sinus beat to result in a longer RR interval, which further exacerbated the APD dispersion giving rise to the spontaneous occurrence of ventricular arrhythmias. Slow heart rate-induced PVC alternans was observed in experiments of transgenic LQT2 rabbits under isoproterenol, which was associated with increased APD dispersion and spontaneous occurrence of ventricular arrhythmias, in agreement with the theoretical predictions.

Mechanism for Triggered Waves in Atrial Myocytes

Excitation-contraction coupling in atrial cells is mediated by calcium (Ca) signaling between L-type Ca channels and Ryanodine receptors that occurs mainly at the cell boundary. This unique architecture dictates essential aspects of Ca signaling under both normal and diseased conditions. In this study we apply laser scanning confocal microscopy, along with an experimentally based computational model, to understand the Ca cycling dynamics of an atrial cell subjected to rapid pacing. Our main finding is that when an atrial cell is paced under Ca overload conditions, Ca waves can then nucleate on the cell boundary and propagate to the cell interior. These propagating Ca waves are referred to as "triggered waves" because they are initiated by L-type Ca channel openings during the action potential. These excitations are distinct from spontaneous Ca waves originating from random fluctuations of Ryanodine receptor channels, and which occur after much longer waiting times. Furthermore, we argue that the onset of these triggered waves is a highly nonlinear function of the sarcoplasmic reticulum Ca load. This strong nonlinearity leads to aperiodic response of Ca at rapid pacing rates that is caused by the complex interplay between paced Ca release and triggered waves. We argue further that this feature of atrial cells leads to dynamic instabilities that may underlie atrial arrhythmias. These studies will serve as a starting point to explore the nonlinear dynamics of atrial cells and will yield insights into the trigger and maintenance of atrial fibrillation.

Alternans in atria: Mechanisms and clinical relevance

Atrial fibrillation is the most common sustained arrhythmia and its prevalence is rapidly rising with the aging of the population. Cardiac alternans, defined as cyclic beat-to-beat alternations in contraction force, action potential (AP) duration and intracellular Ca²⁺ release at constant stimulation rate, has been associated with the development of ventricular arrhythmias. Recent clinical data also provide strong evidence that alternans plays a central role in arrhythmogenesis in atria. The aim of this article is to review the mechanisms that are responsible for repolarization alternans and contribute to the transition from spatially concordant alternans to the more arrhythmogenic spatially discordant alternans in atria.

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.

Nonlinear onset of calcium wave propagation in cardiac cells

Spontaneous calcium (Ca) waves in cardiac myocytes are known to underlie a wide range of cardiac arrhythmias. However, it is not understood which physiological parameters determine the onset of waves. In this study, we explore the relationship between Ca signaling between ion channels and the nucleation of Ca waves. In particular, we apply a master equation approach to analyze the stochastic interaction between neighboring clusters of ryanodine receptor (RyR) channels. Using this analysis, we show that signaling between clusters can be described as a barrier hopping process with exponential sensitivity to system parameters. A consequence of this feature is that the probability that Ca release at a cluster induces release at a neighboring cluster exhibits a sigmoid dependence on the Ca content in the cell. This nonlinearity originates from the regulation of RyR opening due to more than one Ca ion binding site, in conjunction with Ca mediated cooperativity between RyR channels in clusters. We apply a spatially distributed stochastic model of Ca cycling to analyze the physiological consequences of this nonlinearity, and show that it explains the sharp onset of Ca wave nucleation in cardiac cells. Furthermore, we show that this sharp onset can serve as a mechanism for Ca alternans under physiologically relevant conditions. Thus our findings identify the nonlinear features of Ca signaling which potentially underlie the onset of Ca waves and Ca alternans in cardiac cells.

Superresolution modeling of calcium release in the heart

Stable calcium-induced calcium release (CICR) is critical for maintaining normal cellular contraction during cardiac excitation-contraction coupling. The fundamental element of CICR in the heart is the calcium (Ca²⁺) spark, which arises from a cluster of ryanodine receptors (RyR). Opening of these RyR clusters is triggered to produce a local, regenerative release of Ca²⁺ from the sarcoplasmic reticulum (SR). The Ca²⁺ leak out of the SR is an important process for cellular Ca²⁺ management, and it is critically influenced by spark fidelity, i.e., the probability that a spontaneous RyR opening triggers a Ca²⁺ spark. Here, we present a detailed, three-dimensional model of a cardiac Ca²⁺ release unit that incorporates diffusion, intracellular buffering systems, and stochastically gated ion channels. The model exhibits realistic Ca²⁺ sparks and robust Ca²⁺ spark termination across a wide range of geometries and conditions. Furthermore, the model captures the details of Ca²⁺ spark and nonspark-based SR Ca²⁺ leak, and it produces normal excitation-contraction coupling gain. We show that SR luminal Ca²⁺-dependent regulation of the RyR is not critical for spark termination, but it can explain the exponential rise in the SR Ca²⁺ leak-load relationship demonstrated in previous experimental work. Perturbations to subspace dimensions, which have been observed in experimental models of disease, strongly alter Ca²⁺ spark dynamics. In addition, we find that the structure of RyR clusters also influences Ca²⁺ release properties due to variations in inter-RyR coupling via local subspace Ca²⁺ concentration ([Ca²⁺]_ss). These results are illustrated for RyR clusters based on super-resolution stimulated emission depletion microscopy. Finally, we present a believed-novel approach by which the spark fidelity of a RyR cluster can be predicted from structural information of the cluster using the maximum eigenvalue of its adjacency matrix. These results provide critical insights into CICR dynamics in heart, under normal and pathological conditions.

On the Adjacency Matrix of RyR2 Cluster Structures

In the heart, electrical stimulation of cardiac myocytes increases the open probability of sarcolemmal voltage-sensitive Ca2+ channels and flux of Ca2+ into the cells. This increases Ca2+ binding to ligand-gated channels known as ryanodine receptors (RyR2). Their openings cause cell-wide release of Ca2+, which in turn causes muscle contraction and the generation of the mechanical force required to pump blood. In resting myocytes, RyR2s can also open spontaneously giving rise to spatially-confined Ca2+ release events known as "sparks." RyR2s are organized in a lattice to form clusters in the junctional sarcoplasmic reticulum membrane. Our recent work has shown that the spatial arrangement of RyR2s within clusters strongly influences the frequency of Ca2+ sparks. We showed that the probability of a Ca2+ spark occurring when a single RyR2 in the cluster opens spontaneously can be predicted from the precise spatial arrangements of the RyR2s. Thus, "function" follows from "structure." This probability is related to the maximum eigenvalue (λ1) of the adjacency matrix of the RyR2 cluster lattice. In this work, we develop a theoretical framework for understanding this relationship. We present a stochastic contact network model of the Ca2+ spark initiation process. We show that λ1 determines a stability threshold for the formation of Ca2+ sparks in terms of the RyR2 gating transition rates. We recapitulate these results by applying the model to realistic RyR2 cluster structures informed by super-resolution stimulated emission depletion (STED) microscopy. Eigendecomposition of the linearized mean-field contact network model reveals functional subdomains within RyR2 clusters with distinct sensitivities to Ca2+. This work provides novel perspectives on the cardiac Ca2+ release process and a general method for inferring the functional properties of transmembrane receptor clusters from their structure.

Cardiac myocyte alternans in intact heart: Influence of cell-cell coupling and β-adrenergic stimulation

BACKGROUND

Cardiac alternans are proarrhythmic and mechanistically link cardiac mechanical dysfunction and sudden cardiac death. Beat-to-beat alternans occur when beats with large Ca(2+) transients and long action potential duration (APD) alternate with the converse. APD alternans are typically driven by Ca(2+) alternans and sarcoplasmic reticulum (SR) Ca(2+) release alternans. But the effect of intercellular communication via gap junctions (GJ) on alternans in the intact heart remains unknown.

OBJECTIVE

We assessed the effects of cell-to-cell coupling on local alternans in intact Langendorff-perfused mouse hearts, measuring single myocyte [Ca(2+)] alternans synchronization among neighboring cells, and effects of β-adrenergic receptor (β-AR) activation and reduced GJ coupling.

METHODS AND RESULTS

Mouse hearts (C57BL/6) were retrogradely perfused and loaded with Fluo8-AM to record cardiac myocyte [Ca(2+)] in situ with confocal microscopy. Single cell resolution allowed analysis of alternans within the intact organ during alternans induction. Carbenoxolone (25 μM), a GJ inhibitor, significantly increased the occurrence and amplitude of alternans in single cells within the intact heart. Alternans were concordant between neighboring cells throughout the field of view, except transiently during onset. β-AR stimulation only reduced Ca(2+) alternans in tissue that had reduced GJ coupling, matching effects seen in isolated myocytes.

CONCLUSIONS

Ca(2+) alternans among neighboring myocytes is predominantly concordant, likely because of electrical coupling between cells. Consistent with this, partial GJ uncoupling increased propensity and amplitude of Ca(2+) alternans, and made them more sensitive to reversal by β-AR activation, as in isolated myocytes. Electrical coupling between myocytes may thus limit the alternans initiation, but also allow alternans to be more stable once established.

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.

Cellular mechanism of cardiac alternans: an unresolved chicken or egg problem

T-wave alternans, a specific form of cardiac alternans, has been associated with the increased susceptibility to cardiac arrhythmias and sudden cardiac death (SCD). Plenty of evidence has related cardiac alternans at the tissue level to the instability of voltage kinetics or Ca(2+) handling dynamics at the cellular level. However, to date, none of the existing experiments could identify the exact cellular mechanism of cardiac alternans due to the bi-directional coupling between voltage kinetics and Ca(2+) handling dynamics. Either of these systems could be the origin of alternans and the other follows as a secondary change, therefore making the cellular mechanism of alternans a difficult chicken or egg problem. In this context, theoretical analysis combined with experimental techniques provides a possibility to explore this problem. In this review, we will summarize the experimental and theoretical advances in understanding the cellular mechanism of alternans. We focus on the roles of action potential duration (APD) restitution and Ca(2+) handling dynamics in the genesis of alternans and show how the theoretical analysis combined with experimental techniques has provided us a new insight into the cellular mechanism of alternans. We also discuss the possible reasons of increased propensity for alternans in heart failure (HF) and the new possible therapeutic targets. Finally, according to the level of electrophysiological recording techniques and theoretical strategies, we list some critical experimental or theoretical challenges which may help to determine the origin of alternans and to find more effective therapeutic targets in the future.

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.

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.

Trimeric intracellular cation channels and sarcoplasmic/endoplasmic reticulum calcium homeostasis

Trimeric intracellular cation channels (TRIC) represents a novel class of trimeric intracellular cation channels. Two TRIC isoforms have been identified in both the human and the mouse genomes: TRIC-A, a subtype predominantly expressed in the sarcoplasmic reticulum (SR) of muscle cells, and TRIC-B, a ubiquitous subtype expressed in the endoplasmic reticulum (ER) of all tissues. Genetic ablation of either TRIC-A or TRIC-B leads to compromised K(+) permeation and Ca(2+) release across the SR/ER membrane, supporting the hypothesis that TRIC channels provide a counter balancing K(+) flux that reduces SR/ER membrane depolarization for maintenance of the electrochemical gradient that drives SR/ER Ca(2+) release. TRIC-A and TRIC-B seem to have differential functions in Ca(2+) signaling in excitable and nonexcitable cells. Tric-a(-/-) mice display defective Ca(2+) sparks and spontaneous transient outward currents in arterial smooth muscle and develop hypertension, in addition to skeletal muscle dysfunction. Knockout of TRIC-B results in abnormal IP3 receptor-mediated Ca(2+) release in airway epithelial cells, respiratory defects, and neonatal lethality. Double knockout mice lacking both TRIC-A and TRIC-B show embryonic lethality as a result of cardiac arrest. Such an aggravated lethality indicates that TRIC-A and TRIC-B share complementary physiological functions in Ca(2+) signaling in embryonic cardiomyocytes. Tric-a(-/-) and Tric-b(+/-) mice are viable and susceptible to stress-induced heart failure. Recent evidence suggests that TRIC-A directly modulates the function of the cardiac ryanodine receptor 2 Ca(2+) release channel, which in turn controls store-overload-induced Ca(2+) release from the SR. Thus, the TRIC channels, in addition to providing a countercurrent for SR/ER Ca(2+) release, may also function as accessory proteins that directly modulate the ryanodine receptor/IP3 receptor channel functions.

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.

Decoding myocardial Ca²⁺ signals across multiple spatial scales: a role for sensitivity analysis

Numerous studies have employed mathematical modeling to quantitatively understand release of Ca(2+) from the sarcoplasmic reticulum (SR) in the heart. Models have been used to investigate physiologically important phenomena such as triggering of SR Ca(2+) release by Ca(2+) entry across the cell membrane and spontaneous leak of Ca(2+) from the SR in quiescent heart cells. In this review we summarize studies that have modeled myocardial Ca(2+) at different spatial scales: the sub-cellular level, the cellular level, and the multicellular level. We discuss each category of models from the standpoint of parameter sensitivity analysis, a common simulation procedure that can generate quantitative, comprehensive predictions about how changes in conditions influence model output. We propose that this is a useful perspective for conceptualizing models, in part because a sensitivity analysis requires the investigator to define the relevant parameters and model outputs. This procedure therefore helps to illustrate the capabilities and limitations of each model. We further suggest that in future studies, sensitivity analyses will aid in simplifying complex models and in suggesting experiments to differentiate between competing models built with different assumptions. We conclude with a discussion of unresolved questions that are likely to be addressed over the next several years.

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

Intracellular calcium (Ca) alternans in cardiac myocytes have been shown in many experimental studies, and the mechanisms remain incompletely understood. We recently developed a "3R theory" that links Ca sparks to whole cell Ca alternans through three critical properties: randomness of Ca sparks; recruitment of a Ca spark by neighboring Ca sparks; and refractoriness of Ca release units. In this study, we used computer simulation of a physiologically detailed mathematical model of a ventricular myocyte couplon network to study how sarcoplasmic reticulum (SR) Ca load and other physiological parameters, such as ryanodine receptor sensitivity, SR uptake rate, Na-Ca exchange strength, and Ca buffer levels affect Ca alternans in the context of 3R theory. We developed a method to calculate the parameters used in the 3R theory (i.e., the primary spark rate and the recruitment rate) from the physiologically detailed Ca cycling model and paced the model periodically to elicit Ca alternans. We show that alternans only occurs for an intermediate range of the SR Ca load, and the underlying mechanism can be explained via its effects on the 3Rs. Furthermore, we show that altering the physiological parameters not only directly changes the 3Rs but also alters the SR Ca load, having an indirect effect on the 3Rs as well. Therefore, our present study links the SR Ca load and other physiological parameters to whole cell Ca alternans through the framework of the 3R theory, providing a general mechanistic understanding of Ca alternans in ventricular myocytes.

Mechanisms by which cytoplasmic calcium wave propagation and alternans are generated in cardiac atrial myocytes lacking T-tubules-insights from a simulation study

This study investigated the mechanisms underlying the propagation of cytoplasmic calcium waves and the genesis of systolic Ca(2+) alternans in cardiac myocytes lacking transverse tubules (t-tubules). These correspond to atrial cells of either small mammals or large mammals that have lost their t-tubules due to disease-induced structural remodeling (e.g., atrial fibrillation). A mathematical model was developed for a cluster of ryanodine receptors distributed on the cross section of a cell that was divided into 13 elements with a spatial resolution of 2 μm. Due to the absence of t-tubules, L-type Ca(2+) channels were only located in the peripheral elements close to the cell-membrane surface and produced Ca(2+) signals that propagated toward central elements by triggering successive Ca(2+)-induced Ca(2+) release (CICR) via Ca(2+) diffusion between adjacent elements. Under control conditions, the Ca(2+) signals did not fully propagate to the central region of the cell. However, with modulation of several factors responsible for Ca(2+) handling, such as the L-type Ca(2+) channels (Ca(2+) influx), SERCA pumps (sarcoplasmic reticulum (SR) Ca(2+) uptake), and ryanodine receptors (SR Ca(2+) release), Ca(2+) wave propagation to the center of the cell could occur. These simulation results are consistent with previous experimental data from atrial cells of small mammals. The model further reveals that spatially functional heterogeneity in Ca(2+) diffusion within the cell produced a steep relationship between the SR Ca(2+) content and the cytoplasmic Ca(2+) concentration. This played an important role in the genesis of Ca(2+) alternans that were more obvious in central than in peripheral elements. Possible association between the occurrence of Ca(2+) alternans and the model parameters of Ca(2+) handling was comprehensively explored in a wide range of one- and two-parameter spaces. In addition, the model revealed a spontaneous second Ca(2+) release in response to a single voltage stimulus pulse with SR Ca(2+) overloading and augmented Ca(2+) influx. This study provides what to our knowledge are new insights into the genesis of Ca(2+) alternans and spontaneous second Ca(2+) release in cardiac myocytes that lack t-tubules.

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.

T-wave alternans and arrhythmogenesis in cardiac diseases

T-wave alternans, a manifestation of repolarization alternans at the cellular level, is associated with lethal cardiac arrhythmias and sudden cardiac death. At the cellular level, several mechanisms can produce repolarization alternans, including: (1) electrical restitution resulting from collective ion channel recovery, which usually occurs at fast heart rates but can also occur at normal heart rates when action potential is prolonged resulting in a short diastolic interval; (2) the transient outward current, which tends to occur at normal or slow heart rates; (3) the dynamics of early after depolarizations, which tends to occur during bradycardia; and (4) intracellular calcium cycling alternans through its interaction with membrane voltage. In this review, we summarize the cellular mechanisms of alternans arising from these different mechanisms, and discuss their roles in arrhythmogenesis in the setting of cardiac disease.

Ca2+ alternans in a cardiac myocyte model that uses moment equations to represent heterogeneous junctional SR Ca2+

Multiscale whole-cell models that accurately represent local control of Ca2+-induced Ca2+ release in cardiac myocytes can reproduce high-gain Ca2+ release that is graded with changes in membrane potential. Using a recently introduced formalism that represents heterogeneous local Ca2+ using moment equations, we present a model of cardiac myocyte Ca2+ cycling that exhibits alternating sarcoplasmic reticulum (SR) Ca2+ release when periodically stimulated by depolarizing voltage pulses. The model predicts that the distribution of junctional SR [Ca2+] across a large population of Ca2+ release units is distinct on alternating cycles. Load-release and release-uptake functions computed from this model give insight into how Ca2+ fluxes and stimulation frequency combine to determine the presence or absence of Ca2+ alternans. Our results show that the conditions for the onset of Ca2+ alternans cannot be explained solely by the steepness of the load-release function, but that changes in the release-uptake process also play an important role. We analyze the effect of the junctional SR refilling time constant on Ca2+ alternans and conclude that physiologically realistic models of defective Ca2+ cycling must represent the dynamics of heterogeneous junctional SR [Ca2+] without assuming rapid equilibration of junctional and network SR [Ca2+].

Models of cardiac excitation-contraction coupling in ventricular myocytes

Mathematical and computational modeling of cardiac excitation-contraction coupling has produced considerable insights into how the heart muscle contracts. With the increase in biophysical and physiological data available, the modeling has become more sophisticated with investigations spanning in scale from molecular components to whole cells. These modeling efforts have provided insight into cardiac excitation-contraction coupling that advanced and complemented experimental studies. One goal is to extend these detailed cellular models to model the whole heart. While this has been done with mechanical and electrophysiological models, the complexity and fast time course of calcium dynamics have made inclusion of detailed calcium dynamics in whole heart models impractical. Novel methods such as the probability density approach and moment closure technique which increase computational efficiency might make this tractable.

Spark-induced sparks as a mechanism of intracellular calcium alternans in cardiac myocytes

RATIONALE

Intracellular calcium (Ca) alternans has been widely studied in cardiac myocytes and tissue, yet the underlying mechanism remains controversial.

OBJECTIVE

In this study, we used computational modeling and simulation to study how randomly occurring Ca sparks interact collectively to result in whole-cell Ca alternans.

METHODS AND RESULTS

We developed a spatially distributed intracellular Ca cycling model in which Ca release units (CRUs) are locally coupled by Ca diffusion throughout the myoplasm and sarcoplasmic reticulum (SR) network. Ca sparks occur randomly in the CRU network when periodically paced with a clamped voltage waveform, but Ca alternans develops as the pacing speeds up. Combining computational simulation with theoretical analysis, we show that Ca alternans emerges as a collective behavior of Ca sparks, determined by 3 critical properties of the CRU network from which Ca sparks arise: "randomness" (of Ca spark activation), "refractoriness" (of a CRU after a Ca spark), and "recruitment" (Ca sparks inducing Ca sparks in adjacent CRUs). We also show that the steep nonlinear relationship between fractional SR Ca release and SR Ca load arises naturally as a collective behavior of Ca sparks, and Ca alternans can occur even when SR Ca is held constant.

CONCLUSIONS

We present a general theory for the mechanisms of intracellular Ca alternans, which mechanistically links Ca sparks to whole-cell Ca alternans, and is applicable to Ca alternans in both physiological and pathophysiological conditions.

Mathematical models of the electrical action potential of Purkinje fibre cells

Early development of ionic models for cardiac myocytes, from the pioneering modification of the Hodgkin-Huxley giant squid axon model by Noble to the iconic DiFrancesco-Noble model integrating voltage-gated ionic currents, ion pumps and exchangers, Ca(2+) sequestration and Ca(2+)-induced Ca(2+) release, provided a general description for a mammalian Purkinje fibre (PF) and the framework for modern cardiac models. In the past two decades, development has focused on tissue-specific models with an emphasis on the sino-atrial (SA) node, atria and ventricles, while the PFs have largely been neglected. However, achieving the ultimate goal of creating a virtual human heart will require detailed models of all distinctive regions of the cardiac conduction system, including the PFs, which play an important role in conducting cardiac excitation and ensuring the synchronized timing and sequencing of ventricular contraction. In this paper, we present details of our newly developed model for the human PF cell including validation against experimental data. Ionic mechanisms underlying the heterogeneity between the PF and ventricular action potentials in humans and other species are analysed. The newly developed PF cell model adds a new member to the family of human cardiac cell models developed previously for the SA node, atrial and ventricular cells, which can be incorporated into an anatomical model of the human heart with details of its electrophysiological heterogeneity and anatomical complexity.

Modelling sarcoplasmic reticulum calcium ATPase and its regulation in cardiac myocytes

When developing large-scale mathematical models of physiology, some reduction in complexity is necessarily required to maintain computational efficiency. A prime example of such an intricate cell is the cardiac myocyte. For the predictive power of the cardiomyocyte models, it is vital to accurately describe the calcium transport mechanisms, since they essentially link the electrical activation to contractility. The removal of calcium from the cytoplasm takes place mainly by the Na(+)/Ca(2+) exchanger, and the sarcoplasmic reticulum Ca(2+) ATPase (SERCA). In the present study, we review the properties of SERCA, its frequency-dependent and beta-adrenergic regulation, and the approaches of mathematical modelling that have been used to investigate its function. Furthermore, we present novel theoretical considerations that might prove useful for the elucidation of the role of SERCA in cardiac function, achieving a reduction in model complexity, but at the same time retaining the central aspects of its function. Our results indicate that to faithfully predict the physiological properties of SERCA, we should take into account the calcium-buffering effect and reversible function of the pump. This 'uncomplicated' modelling approach could be useful to other similar transport mechanisms as well.

Cardiac alternans induced by fibroblast-myocyte coupling: mechanistic insights from computational models

Recent experimental studies have shown that fibroblasts can electrotonically couple to myocytes via gap junctions. In this study, we investigated how this coupling affects action potential and intracellular calcium (Ca(i)) cycling dynamics in simulated fibroblast-myocyte pairs and in two-dimensional tissue with random fibroblast insertions. We show that a fibroblast coupled with a myocyte generates a gap junction current flowing to the myocyte with two main components: an early pulse of transient outward current, similar to the fast transient outward current, and a later background current during the repolarizing phase. Depending on the relative prominence of the two components, fibroblast-myoycte coupling can 1) prolong or shorten action potential duration (APD), 2) promote or suppress APD alternans due to steep APD restitution (voltage driven) and also result in a novel mechanism of APD alternans at slow heart rates, 3) promote Ca(i)-driven alternans and electromechanically discordant alternans, and 4) promote spatially discordant alternans by two mechanisms: by altering conduction velocity restitution and by heterogeneous fibroblast distribution causing electromechanically concordant and discordant alternans in different regions of the tissue. Thus, through their coupling with myocytes, fibroblasts alter repolarization and Ca(i) cycling alternans at both the cellular and tissue scales, which may play important roles in arrhythmogenesis in diseased cardiac tissue with fibrosis.

The effects of membrane potential, SR Ca2+ content and RyR responsiveness on systolic Ca2+ alternans in rat ventricular myocytes

Previous work has shown that small depolarizing pulses produce a beat to beat alternation in the amplitude of the systolic Ca(2+) transient in ventricular myocytes. The aim of the present work was to investigate the role of changes of SR Ca(2+) content and L-type Ca(2+) current in this alternans. As the amplitude of the depolarizing pulse was increased from 10 to 30 mV the magnitude of alternans decreased. Confocal linescan studies showed that this was accompanied by an increase in the number of sites from which Ca(2+) waves propagated. A sudden decrease in the depolarisation amplitude resulted in three classes of behaviour: (1) a gradual decrease in Ca(2+) transient amplitude before alternans developed accompanied by a loss of SR Ca(2+), (2) a gradual increase in Ca(2+) transient amplitude before alternans accompanied by a gain of SR Ca(2+), and (3) immediate development of alternans with no change of SR content. We conclude that alternans develops if the combination of decreased opening of L-type channels and change of SR Ca(2+) content results in spatially fragmented release from the SR as long as there is sufficient Ca(2+) in the SR to sustain wave propagation. Potentiation of the opening of the ryanodine receptor (RyR) by low concentrations of caffeine (100 microm) abolished alternans for a few pulses but the alternans then redeveloped once SR Ca(2+) content fell to the new threshold for wave propagation. Finally we show evidence that inhibiting L-type Ca(2+) current with 200 mum Cd(2+) produces alternans by means of a similar fragmentation of the Ca(2+) release profile and propagation of mini-waves of Ca(2+) release.